combined.cpp

#include <bits/stdc++.h>

#include <algorithm>
#include <array>

#ifdef _MSC_VER
#include <intrin.h>
#endif

namespace atcoder {

    namespace internal {

        // @param n `0 <= n`
        // @return minimum non-negative `x` s.t. `n <= 2**x`
        int ceil_pow2(int n) {
            int x = 0;
            while ((1U << x) < (unsigned int)(n)) x++;
            return x;
        }

        // @param n `1 <= n`
        // @return minimum non-negative `x` s.t. `(n & (1 << x)) != 0`
        int bsf(unsigned int n) {
#ifdef _MSC_VER
            unsigned long index;
            _BitScanForward(&index, n);
            return index;
#else
            return __builtin_ctz(n);
#endif
        }

    }  // namespace internal

}  // namespace atcoder

#include <utility>

namespace atcoder {

    namespace internal {

        // @param m `1 <= m`
        // @return x mod m
        constexpr long long safe_mod(long long x, long long m) {
            x %= m;
            if (x < 0) x += m;
            return x;
        }

        // Fast moduler by barrett reduction
        // Reference: https://en.wikipedia.org/wiki/Barrett_reduction
        // NOTE: reconsider after Ice Lake
        struct barrett {
            unsigned int _m;
            unsigned long long im;

            // @param m `1 <= m`
            barrett(unsigned int m) : _m(m), im((unsigned long long)(-1) / m + 1) {}

            // @return m
            unsigned int umod() const { return _m; }

            // @param a `0 <= a < m`
            // @param b `0 <= b < m`
            // @return `a * b % m`
            unsigned int mul(unsigned int a, unsigned int b) const {
                // [1] m = 1
                // a = b = im = 0, so okay

                // [2] m >= 2
                // im = ceil(2^64 / m)
                // -> im * m = 2^64 + r (0 <= r < m)
                // let z = a*b = c*m + d (0 <= c, d < m)
                // a*b * im = (c*m + d) * im = c*(im*m) + d*im = c*2^64 + c*r + d*im
                // c*r + d*im < m * m + m * im < m * m + 2^64 + m <= 2^64 + m * (m + 1)
                // < 2^64 * 2
                // ((ab * im) >> 64) == c or c + 1
                unsigned long long z = a;
                z *= b;
#ifdef _MSC_VER
                unsigned long long x;
                _umul128(z, im, &x);
#else
                unsigned long long x =
                    (unsigned long long)(((unsigned __int128)(z)*im) >> 64);
#endif
                unsigned int v = (unsigned int)(z - x * _m);
                if (_m <= v) v += _m;
                return v;
            }
        };

        // @param n `0 <= n`
        // @param m `1 <= m`
        // @return `(x ** n) % m`
        constexpr long long pow_mod_constexpr(long long x, long long n, int m) {
            if (m == 1) return 0;
            unsigned int _m = (unsigned int)(m);
            unsigned long long r = 1;
            unsigned long long y = safe_mod(x, m);
            while (n) {
                if (n & 1) r = (r * y) % _m;
                y = (y * y) % _m;
                n >>= 1;
            }
            return r;
        }

        // Reference:
        // M. Forisek and J. Jancina,
        // Fast Primality Testing for Integers That Fit into a Machine Word
        // @param n `0 <= n`
        constexpr bool is_prime_constexpr(int n) {
            if (n <= 1) return false;
            if (n == 2 || n == 7 || n == 61) return true;
            if (n % 2 == 0) return false;
            long long d = n - 1;
            while (d % 2 == 0) d /= 2;
            for (long long a : {2, 7, 61}) {
                long long t = d;
                long long y = pow_mod_constexpr(a, t, n);
                while (t != n - 1 && y != 1 && y != n - 1) {
                    y = y * y % n;
                    t <<= 1;
                }
                if (y != n - 1 && t % 2 == 0) {
                    return false;
                }
            }
            return true;
        }
        template <int n>
            constexpr bool is_prime = is_prime_constexpr(n);

        // @param b `1 <= b`
        // @return pair(g, x) s.t. g = gcd(a, b), xa = g (mod b), 0 <= x < b/g
        constexpr std::pair<long long, long long> inv_gcd(long long a, long long b) {
            a = safe_mod(a, b);
            if (a == 0) return {b, 0};

            // Contracts:
            // [1] s - m0 * a = 0 (mod b)
            // [2] t - m1 * a = 0 (mod b)
            // [3] s * |m1| + t * |m0| <= b
            long long s = b, t = a;
            long long m0 = 0, m1 = 1;

            while (t) {
                long long u = s / t;
                s -= t * u;
                m0 -= m1 * u;  // |m1 * u| <= |m1| * s <= b

                // [3]:
                // (s - t * u) * |m1| + t * |m0 - m1 * u|
                // <= s * |m1| - t * u * |m1| + t * (|m0| + |m1| * u)
                // = s * |m1| + t * |m0| <= b

                auto tmp = s;
                s = t;
                t = tmp;
                tmp = m0;
                m0 = m1;
                m1 = tmp;
            }
            // by [3]: |m0| <= b/g
            // by g != b: |m0| < b/g
            if (m0 < 0) m0 += b / s;
            return {s, m0};
        }

        // Compile time primitive root
        // @param m must be prime
        // @return primitive root (and minimum in now)
        constexpr int primitive_root_constexpr(int m) {
            if (m == 2) return 1;
            if (m == 167772161) return 3;
            if (m == 469762049) return 3;
            if (m == 754974721) return 11;
            if (m == 998244353) return 3;
            int divs[20] = {};
            divs[0] = 2;
            int cnt = 1;
            int x = (m - 1) / 2;
            while (x % 2 == 0) x /= 2;
            for (int i = 3; (long long)(i)*i <= x; i += 2) {
                if (x % i == 0) {
                    divs[cnt++] = i;
                    while (x % i == 0) {
                        x /= i;
                    }
                }
            }
            if (x > 1) {
                divs[cnt++] = x;
            }
            for (int g = 2;; g++) {
                bool ok = true;
                for (int i = 0; i < cnt; i++) {
                    if (pow_mod_constexpr(g, (m - 1) / divs[i], m) == 1) {
                        ok = false;
                        break;
                    }
                }
                if (ok) return g;
            }
        }
        template <int m>
            constexpr int primitive_root = primitive_root_constexpr(m);

    }  // namespace internal

}  // namespace atcoder

#include <cassert>
#include <numeric>
#include <type_traits>

namespace atcoder {

    namespace internal {

#ifndef _MSC_VER
        template <class T>
            using is_signed_int128 =
            typename std::conditional<std::is_same<T, __int128_t>::value ||
            std::is_same<T, __int128>::value,
            std::true_type, std::false_type>::type;

        template <class T>
            using is_unsigned_int128 =
            typename std::conditional<std::is_same<T, __uint128_t>::value ||
            std::is_same<T, unsigned __int128>::value,
            std::true_type, std::false_type>::type;

        template <class T>
            using make_unsigned_int128 =
            typename std::conditional<std::is_same<T, __int128_t>::value, __uint128_t,
                     unsigned __int128>;

        template <class T>
            using is_integral =
            typename std::conditional<std::is_integral<T>::value ||
            is_signed_int128<T>::value ||
            is_unsigned_int128<T>::value,
            std::true_type, std::false_type>::type;

        template <class T>
            using is_signed_int =
            typename std::conditional<(is_integral<T>::value &&
                    std::is_signed<T>::value) ||
            is_signed_int128<T>::value,
            std::true_type, std::false_type>::type;

        template <class T>
            using is_unsigned_int =
            typename std::conditional<(is_integral<T>::value &&
                    std::is_unsigned<T>::value) ||
            is_unsigned_int128<T>::value,
            std::true_type, std::false_type>::type;

        template <class T>
            using to_unsigned = typename std::conditional<
            is_signed_int128<T>::value, make_unsigned_int128<T>,
            typename std::conditional<std::is_signed<T>::value, std::make_unsigned<T>,
            std::common_type<T>>::type>::type;

#else

        template <class T>
            using is_integral = typename std::is_integral<T>;

        template <class T>
            using is_signed_int =
            typename std::conditional<is_integral<T>::value && std::is_signed<T>::value,
                     std::true_type, std::false_type>::type;

        template <class T>
            using is_unsigned_int =
            typename std::conditional<is_integral<T>::value &&
            std::is_unsigned<T>::value,
            std::true_type, std::false_type>::type;

        template <class T>
            using to_unsigned =
            typename std::conditional<is_signed_int<T>::value, std::make_unsigned<T>,
                     std::common_type<T>>::type;

#endif

        template <class T>
            using is_signed_int_t = std::enable_if_t<is_signed_int<T>::value>;

        template <class T>
            using is_unsigned_int_t = std::enable_if_t<is_unsigned_int<T>::value>;

        template <class T>
            using to_unsigned_t = typename to_unsigned<T>::type;

    }  // namespace internal

}  // namespace atcoder

#include <cassert>
#include <numeric>
#include <type_traits>

#ifdef _MSC_VER
#include <intrin.h>
#endif

namespace atcoder {

    namespace internal {

        struct modint_base {};
        struct static_modint_base : modint_base {};

        template <class T>
            using is_modint = std::is_base_of<modint_base, T>;
        template <class T>
            using is_modint_t = std::enable_if_t<is_modint<T>::value>;

    }  // namespace internal

    template <int m, std::enable_if_t<(1 <= m)>* = nullptr>
        struct static_modint : internal::static_modint_base {
            using mint = static_modint;

            public:
            static constexpr int mod() { return m; }
            static mint raw(int v) {
                mint x;
                x._v = v;
                return x;
            }

            static_modint() : _v(0) {}
            template <class T, internal::is_signed_int_t<T>* = nullptr>
                static_modint(T v) {
                    long long x = (long long)(v % (long long)(umod()));
                    if (x < 0) x += umod();
                    _v = (unsigned int)(x);
                }
            template <class T, internal::is_unsigned_int_t<T>* = nullptr>
                static_modint(T v) {
                    _v = (unsigned int)(v % umod());
                }
            static_modint(bool v) { _v = ((unsigned int)(v) % umod()); }

            unsigned int val() const { return _v; }

            mint& operator++() {
                _v++;
                if (_v == umod()) _v = 0;
                return *this;
            }
            mint& operator--() {
                if (_v == 0) _v = umod();
                _v--;
                return *this;
            }
            mint operator++(int) {
                mint result = *this;
                ++*this;
                return result;
            }
            mint operator--(int) {
                mint result = *this;
                --*this;
                return result;
            }

            mint& operator+=(const mint& rhs) {
                _v += rhs._v;
                if (_v >= umod()) _v -= umod();
                return *this;
            }
            mint& operator-=(const mint& rhs) {
                _v -= rhs._v;
                if (_v >= umod()) _v += umod();
                return *this;
            }
            mint& operator*=(const mint& rhs) {
                unsigned long long z = _v;
                z *= rhs._v;
                _v = (unsigned int)(z % umod());
                return *this;
            }
            mint& operator/=(const mint& rhs) { return *this = *this * rhs.inv(); }

            mint operator+() const { return *this; }
            mint operator-() const { return mint() - *this; }

            mint pow(long long n) const {
                assert(0 <= n);
                mint x = *this, r = 1;
                while (n) {
                    if (n & 1) r *= x;
                    x *= x;
                    n >>= 1;
                }
                return r;
            }
            mint inv() const {
                if (prime) {
                    assert(_v);
                    return pow(umod() - 2);
                } else {
                    auto eg = internal::inv_gcd(_v, m);
                    assert(eg.first == 1);
                    return eg.second;
                }
            }

            friend mint operator+(const mint& lhs, const mint& rhs) {
                return mint(lhs) += rhs;
            }
            friend mint operator-(const mint& lhs, const mint& rhs) {
                return mint(lhs) -= rhs;
            }
            friend mint operator*(const mint& lhs, const mint& rhs) {
                return mint(lhs) *= rhs;
            }
            friend mint operator/(const mint& lhs, const mint& rhs) {
                return mint(lhs) /= rhs;
            }
            friend bool operator==(const mint& lhs, const mint& rhs) {
                return lhs._v == rhs._v;
            }
            friend bool operator!=(const mint& lhs, const mint& rhs) {
                return lhs._v != rhs._v;
            }

            private:
            unsigned int _v;
            static constexpr unsigned int umod() { return m; }
            static constexpr bool prime = internal::is_prime<m>;
        };

    template <int id>
        struct dynamic_modint : internal::modint_base {
            using mint = dynamic_modint;

            public:
            static int mod() { return (int)(bt.umod()); }
            static void set_mod(int m) {
                assert(1 <= m);
                bt = internal::barrett(m);
            }
            static mint raw(int v) {
                mint x;
                x._v = v;
                return x;
            }

            dynamic_modint() : _v(0) {}
            template <class T, internal::is_signed_int_t<T>* = nullptr>
                dynamic_modint(T v) {
                    long long x = (long long)(v % (long long)(mod()));
                    if (x < 0) x += mod();
                    _v = (unsigned int)(x);
                }
            template <class T, internal::is_unsigned_int_t<T>* = nullptr>
                dynamic_modint(T v) {
                    _v = (unsigned int)(v % mod());
                }
            dynamic_modint(bool v) { _v = ((unsigned int)(v) % mod()); }

            unsigned int val() const { return _v; }

            mint& operator++() {
                _v++;
                if (_v == umod()) _v = 0;
                return *this;
            }
            mint& operator--() {
                if (_v == 0) _v = umod();
                _v--;
                return *this;
            }
            mint operator++(int) {
                mint result = *this;
                ++*this;
                return result;
            }
            mint operator--(int) {
                mint result = *this;
                --*this;
                return result;
            }

            mint& operator+=(const mint& rhs) {
                _v += rhs._v;
                if (_v >= umod()) _v -= umod();
                return *this;
            }
            mint& operator-=(const mint& rhs) {
                _v += mod() - rhs._v;
                if (_v >= umod()) _v -= umod();
                return *this;
            }
            mint& operator*=(const mint& rhs) {
                _v = bt.mul(_v, rhs._v);
                return *this;
            }
            mint& operator/=(const mint& rhs) { return *this = *this * rhs.inv(); }

            mint operator+() const { return *this; }
            mint operator-() const { return mint() - *this; }

            mint pow(long long n) const {
                assert(0 <= n);
                mint x = *this, r = 1;
                while (n) {
                    if (n & 1) r *= x;
                    x *= x;
                    n >>= 1;
                }
                return r;
            }
            mint inv() const {
                auto eg = internal::inv_gcd(_v, mod());
                assert(eg.first == 1);
                return eg.second;
            }

            friend mint operator+(const mint& lhs, const mint& rhs) {
                return mint(lhs) += rhs;
            }
            friend mint operator-(const mint& lhs, const mint& rhs) {
                return mint(lhs) -= rhs;
            }
            friend mint operator*(const mint& lhs, const mint& rhs) {
                return mint(lhs) *= rhs;
            }
            friend mint operator/(const mint& lhs, const mint& rhs) {
                return mint(lhs) /= rhs;
            }
            friend bool operator==(const mint& lhs, const mint& rhs) {
                return lhs._v == rhs._v;
            }
            friend bool operator!=(const mint& lhs, const mint& rhs) {
                return lhs._v != rhs._v;
            }

            private:
            unsigned int _v;
            static internal::barrett bt;
            static unsigned int umod() { return bt.umod(); }
        };
    template <int id>
        internal::barrett dynamic_modint<id>::bt = 998244353;

    using modint998244353 = static_modint<998244353>;
    using modint1000000007 = static_modint<1000000007>;
    using modint = dynamic_modint<-1>;

    namespace internal {

        template <class T>
            using is_static_modint = std::is_base_of<internal::static_modint_base, T>;

        template <class T>
            using is_static_modint_t = std::enable_if_t<is_static_modint<T>::value>;

        template <class>
            struct is_dynamic_modint : public std::false_type {};
        template <int id>
            struct is_dynamic_modint<dynamic_modint<id>> : public std::true_type {};

        template <class T>
            using is_dynamic_modint_t = std::enable_if_t<is_dynamic_modint<T>::value>;

    }  // namespace internal

}  // namespace atcoder

#include <cassert>
#include <type_traits>
#include <vector>

namespace atcoder {

    namespace internal {

        template <class mint, internal::is_static_modint_t<mint>* = nullptr>
            void butterfly(std::vector<mint>& a) {
                static constexpr int g = internal::primitive_root<mint::mod()>;
                int n = int(a.size());
                int h = internal::ceil_pow2(n);

                static bool first = true;
                static mint sum_e[30];  // sum_e[i] = ies[0] * ... * ies[i - 1] * es[i]
                if (first) {
                    first = false;
                    mint es[30], ies[30];  // es[i]^(2^(2+i)) == 1
                    int cnt2 = bsf(mint::mod() - 1);
                    mint e = mint(g).pow((mint::mod() - 1) >> cnt2), ie = e.inv();
                    for (int i = cnt2; i >= 2; i--) {
                        // e^(2^i) == 1
                        es[i - 2] = e;
                        ies[i - 2] = ie;
                        e *= e;
                        ie *= ie;
                    }
                    mint now = 1;
                    for (int i = 0; i < cnt2 - 2; i++) {
                        sum_e[i] = es[i] * now;
                        now *= ies[i];
                    }
                }
                for (int ph = 1; ph <= h; ph++) {
                    int w = 1 << (ph - 1), p = 1 << (h - ph);
                    mint now = 1;
                    for (int s = 0; s < w; s++) {
                        int offset = s << (h - ph + 1);
                        for (int i = 0; i < p; i++) {
                            auto l = a[i + offset];
                            auto r = a[i + offset + p] * now;
                            a[i + offset] = l + r;
                            a[i + offset + p] = l - r;
                        }
                        now *= sum_e[bsf(~(unsigned int)(s))];
                    }
                }
            }

        template <class mint, internal::is_static_modint_t<mint>* = nullptr>
            void butterfly_inv(std::vector<mint>& a) {
                static constexpr int g = internal::primitive_root<mint::mod()>;
                int n = int(a.size());
                int h = internal::ceil_pow2(n);

                static bool first = true;
                static mint sum_ie[30];  // sum_ie[i] = es[0] * ... * es[i - 1] * ies[i]
                if (first) {
                    first = false;
                    mint es[30], ies[30];  // es[i]^(2^(2+i)) == 1
                    int cnt2 = bsf(mint::mod() - 1);
                    mint e = mint(g).pow((mint::mod() - 1) >> cnt2), ie = e.inv();
                    for (int i = cnt2; i >= 2; i--) {
                        // e^(2^i) == 1
                        es[i - 2] = e;
                        ies[i - 2] = ie;
                        e *= e;
                        ie *= ie;
                    }
                    mint now = 1;
                    for (int i = 0; i < cnt2 - 2; i++) {
                        sum_ie[i] = ies[i] * now;
                        now *= es[i];
                    }
                }

                for (int ph = h; ph >= 1; ph--) {
                    int w = 1 << (ph - 1), p = 1 << (h - ph);
                    mint inow = 1;
                    for (int s = 0; s < w; s++) {
                        int offset = s << (h - ph + 1);
                        for (int i = 0; i < p; i++) {
                            auto l = a[i + offset];
                            auto r = a[i + offset + p];
                            a[i + offset] = l + r;
                            a[i + offset + p] =
                                (unsigned long long)(mint::mod() + l.val() - r.val()) *
                                inow.val();
                        }
                        inow *= sum_ie[bsf(~(unsigned int)(s))];
                    }
                }
            }

    }  // namespace internal

    template <class mint, internal::is_static_modint_t<mint>* = nullptr>
        std::vector<mint> convolution(std::vector<mint> a, std::vector<mint> b) {
            int n = int(a.size()), m = int(b.size());
            if (!n || !m) return {};
            if (std::min(n, m) <= 60) {
                if (n < m) {
                    std::swap(n, m);
                    std::swap(a, b);
                }
                std::vector<mint> ans(n + m - 1);
                for (int i = 0; i < n; i++) {
                    for (int j = 0; j < m; j++) {
                        ans[i + j] += a[i] * b[j];
                    }
                }
                return ans;
            }
            int z = 1 << internal::ceil_pow2(n + m - 1);
            a.resize(z);
            internal::butterfly(a);
            b.resize(z);
            internal::butterfly(b);
            for (int i = 0; i < z; i++) {
                a[i] *= b[i];
            }
            internal::butterfly_inv(a);
            a.resize(n + m - 1);
            mint iz = mint(z).inv();
            for (int i = 0; i < n + m - 1; i++) a[i] *= iz;
            return a;
        }

    template <unsigned int mod = 998244353, class T,
             std::enable_if_t<internal::is_integral<T>::value>* = nullptr>
                 std::vector<T> convolution(const std::vector<T>& a, const std::vector<T>& b) {
                     int n = int(a.size()), m = int(b.size());
                     if (!n || !m) return {};

                     using mint = static_modint<mod>;
                     std::vector<mint> a2(n), b2(m);
                     for (int i = 0; i < n; i++) {
                         a2[i] = mint(a[i]);
                     }
                     for (int i = 0; i < m; i++) {
                         b2[i] = mint(b[i]);
                     }
                     auto c2 = convolution(move(a2), move(b2));
                     std::vector<T> c(n + m - 1);
                     for (int i = 0; i < n + m - 1; i++) {
                         c[i] = c2[i].val();
                     }
                     return c;
                 }

    std::vector<long long> convolution_ll(const std::vector<long long>& a,
            const std::vector<long long>& b) {
        int n = int(a.size()), m = int(b.size());
        if (!n || !m) return {};

        static constexpr unsigned long long MOD1 = 754974721;  // 2^24
        static constexpr unsigned long long MOD2 = 167772161;  // 2^25
        static constexpr unsigned long long MOD3 = 469762049;  // 2^26
        static constexpr unsigned long long M2M3 = MOD2 * MOD3;
        static constexpr unsigned long long M1M3 = MOD1 * MOD3;
        static constexpr unsigned long long M1M2 = MOD1 * MOD2;
        static constexpr unsigned long long M1M2M3 = MOD1 * MOD2 * MOD3;

        static constexpr unsigned long long i1 =
            internal::inv_gcd(MOD2 * MOD3, MOD1).second;
        static constexpr unsigned long long i2 =
            internal::inv_gcd(MOD1 * MOD3, MOD2).second;
        static constexpr unsigned long long i3 =
            internal::inv_gcd(MOD1 * MOD2, MOD3).second;

        auto c1 = convolution<MOD1>(a, b);
        auto c2 = convolution<MOD2>(a, b);
        auto c3 = convolution<MOD3>(a, b);

        std::vector<long long> c(n + m - 1);
        for (int i = 0; i < n + m - 1; i++) {
            unsigned long long x = 0;
            x += (c1[i] * i1) % MOD1 * M2M3;
            x += (c2[i] * i2) % MOD2 * M1M3;
            x += (c3[i] * i3) % MOD3 * M1M2;
            // B = 2^63, -B <= x, r(real value) < B
            // (x, x - M, x - 2M, or x - 3M) = r (mod 2B)
            // r = c1[i] (mod MOD1)
            // focus on MOD1
            // r = x, x - M', x - 2M', x - 3M' (M' = M % 2^64) (mod 2B)
            // r = x,
            //     x - M' + (0 or 2B),
            //     x - 2M' + (0, 2B or 4B),
            //     x - 3M' + (0, 2B, 4B or 6B) (without mod!)
            // (r - x) = 0, (0)
            //           - M' + (0 or 2B), (1)
            //           -2M' + (0 or 2B or 4B), (2)
            //           -3M' + (0 or 2B or 4B or 6B) (3) (mod MOD1)
            // we checked that
            //   ((1) mod MOD1) mod 5 = 2
            //   ((2) mod MOD1) mod 5 = 3
            //   ((3) mod MOD1) mod 5 = 4
            long long diff =
                c1[i] - internal::safe_mod((long long)(x), (long long)(MOD1));
            if (diff < 0) diff += MOD1;
            static constexpr unsigned long long offset[5] = {
                0, 0, M1M2M3, 2 * M1M2M3, 3 * M1M2M3};
            x -= offset[diff % 5];
            c[i] = x;
        }

        return c;
    }

}  // namespace atcoder

#include <algorithm>
#include <cassert>
#include <vector>

namespace atcoder {

    // Implement (union by size) + (path compression)
    // Reference:
    // Zvi Galil and Giuseppe F. Italiano,
    // Data structures and algorithms for disjoint set union problems
    struct dsu {
        public:
            dsu() : _n(0) {}
            dsu(int n) : _n(n), parent_or_size(n, -1) {}

            int merge(int a, int b) {
                assert(0 <= a && a < _n);
                assert(0 <= b && b < _n);
                int x = leader(a), y = leader(b);
                if (x == y) return x;
                if (-parent_or_size[x] < -parent_or_size[y]) std::swap(x, y);
                parent_or_size[x] += parent_or_size[y];
                parent_or_size[y] = x;
                return x;
            }

            bool same(int a, int b) {
                assert(0 <= a && a < _n);
                assert(0 <= b && b < _n);
                return leader(a) == leader(b);
            }

            int leader(int a) {
                assert(0 <= a && a < _n);
                if (parent_or_size[a] < 0) return a;
                return parent_or_size[a] = leader(parent_or_size[a]);
            }

            int size(int a) {
                assert(0 <= a && a < _n);
                return -parent_or_size[leader(a)];
            }

            std::vector<std::vector<int>> groups() {
                std::vector<int> leader_buf(_n), group_size(_n);
                for (int i = 0; i < _n; i++) {
                    leader_buf[i] = leader(i);
                    group_size[leader_buf[i]]++;
                }
                std::vector<std::vector<int>> result(_n);
                for (int i = 0; i < _n; i++) {
                    result[i].reserve(group_size[i]);
                }
                for (int i = 0; i < _n; i++) {
                    result[leader_buf[i]].push_back(i);
                }
                result.erase(std::remove_if(
                            result.begin(), result.end(),
                            [&](const std::vector<int>& v) { return v.empty(); }),
                        result.end());
                return result;
            }

        private:
            int _n;
            // root node: -1 * component size
            // otherwise: parent
            std::vector<int> parent_or_size;
    };

}  // namespace atcoder

#include <cassert>
#include <vector>

namespace atcoder {

    // Reference: https://en.wikipedia.org/wiki/Fenwick_tree
    template <class T>
        struct fenwick_tree {
            using U = internal::to_unsigned_t<T>;

            public:
            fenwick_tree() : _n(0) {}
            fenwick_tree(int n) : _n(n), data(n) {}

            void add(int p, T x) {
                assert(0 <= p && p < _n);
                p++;
                while (p <= _n) {
                    data[p - 1] += U(x);
                    p += p & -p;
                }
            }

            T sum(int l, int r) {
                assert(0 <= l && l <= r && r <= _n);
                return sum(r) - sum(l);
            }

            private:
            int _n;
            std::vector<U> data;

            U sum(int r) {
                U s = 0;
                while (r > 0) {
                    s += data[r - 1];
                    r -= r & -r;
                }
                return s;
            }
        };

}  // namespace atcoder

#include <algorithm>
#include <cassert>
#include <iostream>
#include <vector>
namespace atcoder {

    template <class S, S (*op)(S, S), S (*e)(), class F, S (*mapping)(F, S),
             F (*composition)(F, F), F (*id)()>
                 struct lazy_segtree {
                     public:
                         lazy_segtree() : lazy_segtree(0) {}
                         lazy_segtree(int n) : lazy_segtree(std::vector<S>(n, e())) {}
                         lazy_segtree(const std::vector<S>& v) : _n(int(v.size())) {
                             log = internal::ceil_pow2(_n);
                             size = 1 << log;
                             d = std::vector<S>(2 * size, e());
                             lz = std::vector<F>(size, id());
                             for (int i = 0; i < _n; i++) d[size + i] = v[i];
                             for (int i = size - 1; i >= 1; i--) {
                                 update(i);
                             }
                         }

                         void set(int p, S x) {
                             assert(0 <= p && p < _n);
                             p += size;
                             for (int i = log; i >= 1; i--) push(p >> i);
                             d[p] = x;
                             for (int i = 1; i <= log; i++) update(p >> i);
                         }

                         S get(int p) {
                             assert(0 <= p && p < _n);
                             p += size;
                             for (int i = log; i >= 1; i--) push(p >> i);
                             return d[p];
                         }

                         S prod(int l, int r) {
                             assert(0 <= l && l <= r && r <= _n);
                             if (l == r) return e();

                             l += size;
                             r += size;

                             for (int i = log; i >= 1; i--) {
                                 if (((l >> i) << i) != l) push(l >> i);
                                 if (((r >> i) << i) != r) push(r >> i);
                             }

                             S sml = e(), smr = e();
                             while (l < r) {
                                 if (l & 1) sml = op(sml, d[l++]);
                                 if (r & 1) smr = op(d[--r], smr);
                                 l >>= 1;
                                 r >>= 1;
                             }

                             return op(sml, smr);
                         }

                         S all_prod() { return d[1]; }

                         void apply(int p, F f) {
                             assert(0 <= p && p < _n);
                             p += size;
                             for (int i = log; i >= 1; i--) push(p >> i);
                             d[p] = mapping(f, d[p]);
                             for (int i = 1; i <= log; i++) update(p >> i);
                         }
                         void apply(int l, int r, F f) {
                             assert(0 <= l && l <= r && r <= _n);
                             if (l == r) return;

                             l += size;
                             r += size;

                             for (int i = log; i >= 1; i--) {
                                 if (((l >> i) << i) != l) push(l >> i);
                                 if (((r >> i) << i) != r) push((r - 1) >> i);
                             }

                             {
                                 int l2 = l, r2 = r;
                                 while (l < r) {
                                     if (l & 1) all_apply(l++, f);
                                     if (r & 1) all_apply(--r, f);
                                     l >>= 1;
                                     r >>= 1;
                                 }
                                 l = l2;
                                 r = r2;
                             }

                             for (int i = 1; i <= log; i++) {
                                 if (((l >> i) << i) != l) update(l >> i);
                                 if (((r >> i) << i) != r) update((r - 1) >> i);
                             }
                         }

                         template <bool (*g)(S)>
                             int max_right(int l) {
                                 return max_right(l, [](S x) { return g(x); });
                             }
                         template <class G>
                             int max_right(int l, G g) {
                                 assert(0 <= l && l <= _n);
                                 assert(g(e()));
                                 if (l == _n) return _n;
                                 l += size;
                                 for (int i = log; i >= 1; i--) push(l >> i);
                                 S sm = e();
                                 do {
                                     while (l % 2 == 0) l >>= 1;
                                     if (!g(op(sm, d[l]))) {
                                         while (l < size) {
                                             push(l);
                                             l = (2 * l);
                                             if (g(op(sm, d[l]))) {
                                                 sm = op(sm, d[l]);
                                                 l++;
                                             }
                                         }
                                         return l - size;
                                     }
                                     sm = op(sm, d[l]);
                                     l++;
                                 } while ((l & -l) != l);
                                 return _n;
                             }

                         template <bool (*g)(S)>
                             int min_left(int r) {
                                 return min_left(r, [](S x) { return g(x); });
                             }
                         template <class G>
                             int min_left(int r, G g) {
                                 assert(0 <= r && r <= _n);
                                 assert(g(e()));
                                 if (r == 0) return 0;
                                 r += size;
                                 for (int i = log; i >= 1; i--) push((r - 1) >> i);
                                 S sm = e();
                                 do {
                                     r--;
                                     while (r > 1 && (r % 2)) r >>= 1;
                                     if (!g(op(d[r], sm))) {
                                         while (r < size) {
                                             push(r);
                                             r = (2 * r + 1);
                                             if (g(op(d[r], sm))) {
                                                 sm = op(d[r], sm);
                                                 r--;
                                             }
                                         }
                                         return r + 1 - size;
                                     }
                                     sm = op(d[r], sm);
                                 } while ((r & -r) != r);
                                 return 0;
                             }

                     private:
                         int _n, size, log;
                         std::vector<S> d;
                         std::vector<F> lz;

                         void update(int k) { d[k] = op(d[2 * k], d[2 * k + 1]); }
                         void all_apply(int k, F f) {
                             d[k] = mapping(f, d[k]);
                             if (k < size) lz[k] = composition(f, lz[k]);
                         }
                         void push(int k) {
                             all_apply(2 * k, lz[k]);
                             all_apply(2 * k + 1, lz[k]);
                             lz[k] = id();
                         }
                 };

}  // namespace atcoder

#include <algorithm>
#include <cassert>
#include <tuple>
#include <vector>

namespace atcoder {

    long long pow_mod(long long x, long long n, int m) {
        assert(0 <= n && 1 <= m);
        if (m == 1) return 0;
        internal::barrett bt((unsigned int)(m));
        unsigned int r = 1, y = (unsigned int)(internal::safe_mod(x, m));
        while (n) {
            if (n & 1) r = bt.mul(r, y);
            y = bt.mul(y, y);
            n >>= 1;
        }
        return r;
    }

    long long inv_mod(long long x, long long m) {
        assert(1 <= m);
        auto z = internal::inv_gcd(x, m);
        assert(z.first == 1);
        return z.second;
    }

    // (rem, mod)
    std::pair<long long, long long> crt(const std::vector<long long>& r,
            const std::vector<long long>& m) {
        assert(r.size() == m.size());
        int n = int(r.size());
        // Contracts: 0 <= r0 < m0
        long long r0 = 0, m0 = 1;
        for (int i = 0; i < n; i++) {
            assert(1 <= m[i]);
            long long r1 = internal::safe_mod(r[i], m[i]), m1 = m[i];
            if (m0 < m1) {
                std::swap(r0, r1);
                std::swap(m0, m1);
            }
            if (m0 % m1 == 0) {
                if (r0 % m1 != r1) return {0, 0};
                continue;
            }
            // assume: m0 > m1, lcm(m0, m1) >= 2 * max(m0, m1)

            // (r0, m0), (r1, m1) -> (r2, m2 = lcm(m0, m1));
            // r2 % m0 = r0
            // r2 % m1 = r1
            // -> (r0 + x*m0) % m1 = r1
            // -> x*u0*g % (u1*g) = (r1 - r0) (u0*g = m0, u1*g = m1)
            // -> x = (r1 - r0) / g * inv(u0) (mod u1)

            // im = inv(u0) (mod u1) (0 <= im < u1)
            long long g, im;
            std::tie(g, im) = internal::inv_gcd(m0, m1);

            long long u1 = (m1 / g);
            // |r1 - r0| < (m0 + m1) <= lcm(m0, m1)
            if ((r1 - r0) % g) return {0, 0};

            // u1 * u1 <= m1 * m1 / g / g <= m0 * m1 / g = lcm(m0, m1)
            long long x = (r1 - r0) / g % u1 * im % u1;

            // |r0| + |m0 * x|
            // < m0 + m0 * (u1 - 1)
            // = m0 + m0 * m1 / g - m0
            // = lcm(m0, m1)
            r0 += x * m0;
            m0 *= u1;  // -> lcm(m0, m1)
            if (r0 < 0) r0 += m0;
        }
        return {r0, m0};
    }

    long long floor_sum(long long n, long long m, long long a, long long b) {
        long long ans = 0;
        if (a >= m) {
            ans += (n - 1) * n * (a / m) / 2;
            a %= m;
        }
        if (b >= m) {
            ans += n * (b / m);
            b %= m;
        }

        long long y_max = (a * n + b) / m, x_max = (y_max * m - b);
        if (y_max == 0) return ans;
        ans += (n - (x_max + a - 1) / a) * y_max;
        ans += floor_sum(y_max, a, m, (a - x_max % a) % a);
        return ans;
    }

}  // namespace atcoder

#include <algorithm>
#include <vector>

namespace atcoder {

    namespace internal {

        template <class T>
            struct simple_queue {
                std::vector<T> payload;
                int pos = 0;
                void reserve(int n) { payload.reserve(n); }
                int size() const { return int(payload.size()) - pos; }
                bool empty() const { return pos == int(payload.size()); }
                void push(const T& t) { payload.push_back(t); }
                T& front() { return payload[pos]; }
                void clear() {
                    payload.clear();
                    pos = 0;
                }
                void pop() { pos++; }
            };

    }  // namespace internal

}  // namespace atcoder

#include <cassert>
#include <limits>
#include <queue>
#include <vector>

namespace atcoder {

    template <class Cap>
        struct mf_graph {
            public:
                mf_graph() : _n(0) {}
                mf_graph(int n) : _n(n), g(n) {}

                int add_edge(int from, int to, Cap cap) {
                    assert(0 <= from && from < _n);
                    assert(0 <= to && to < _n);
                    assert(0 <= cap);
                    int m = int(pos.size());
                    pos.push_back({from, int(g[from].size())});
                    g[from].push_back(_edge{to, int(g[to].size()), cap});
                    g[to].push_back(_edge{from, int(g[from].size()) - 1, 0});
                    return m;
                }

                struct edge {
                    int from, to;
                    Cap cap, flow;
                };

                edge get_edge(int i) {
                    int m = int(pos.size());
                    assert(0 <= i && i < m);
                    auto _e = g[pos[i].first][pos[i].second];
                    auto _re = g[_e.to][_e.rev];
                    return edge{pos[i].first, _e.to, _e.cap + _re.cap, _re.cap};
                }
                std::vector<edge> edges() {
                    int m = int(pos.size());
                    std::vector<edge> result;
                    for (int i = 0; i < m; i++) {
                        result.push_back(get_edge(i));
                    }
                    return result;
                }
                void change_edge(int i, Cap new_cap, Cap new_flow) {
                    int m = int(pos.size());
                    assert(0 <= i && i < m);
                    assert(0 <= new_flow && new_flow <= new_cap);
                    auto& _e = g[pos[i].first][pos[i].second];
                    auto& _re = g[_e.to][_e.rev];
                    _e.cap = new_cap - new_flow;
                    _re.cap = new_flow;
                }

                Cap flow(int s, int t) {
                    return flow(s, t, std::numeric_limits<Cap>::max());
                }
                Cap flow(int s, int t, Cap flow_limit) {
                    assert(0 <= s && s < _n);
                    assert(0 <= t && t < _n);

                    std::vector<int> level(_n), iter(_n);
                    internal::simple_queue<int> que;

                    auto bfs = [&]() {
                        std::fill(level.begin(), level.end(), -1);
                        level[s] = 0;
                        que.clear();
                        que.push(s);
                        while (!que.empty()) {
                            int v = que.front();
                            que.pop();
                            for (auto e : g[v]) {
                                if (e.cap == 0 || level[e.to] >= 0) continue;
                                level[e.to] = level[v] + 1;
                                if (e.to == t) return;
                                que.push(e.to);
                            }
                        }
                    };
                    auto dfs = [&](auto self, int v, Cap up) {
                        if (v == s) return up;
                        Cap res = 0;
                        int level_v = level[v];
                        for (int& i = iter[v]; i < int(g[v].size()); i++) {
                            _edge& e = g[v][i];
                            if (level_v <= level[e.to] || g[e.to][e.rev].cap == 0) continue;
                            Cap d =
                                self(self, e.to, std::min(up - res, g[e.to][e.rev].cap));
                            if (d <= 0) continue;
                            g[v][i].cap += d;
                            g[e.to][e.rev].cap -= d;
                            res += d;
                            if (res == up) break;
                        }
                        return res;
                    };

                    Cap flow = 0;
                    while (flow < flow_limit) {
                        bfs();
                        if (level[t] == -1) break;
                        std::fill(iter.begin(), iter.end(), 0);
                        while (flow < flow_limit) {
                            Cap f = dfs(dfs, t, flow_limit - flow);
                            if (!f) break;
                            flow += f;
                        }
                    }
                    return flow;
                }

                std::vector<bool> min_cut(int s) {
                    std::vector<bool> visited(_n);
                    internal::simple_queue<int> que;
                    que.push(s);
                    while (!que.empty()) {
                        int p = que.front();
                        que.pop();
                        visited[p] = true;
                        for (auto e : g[p]) {
                            if (e.cap && !visited[e.to]) {
                                visited[e.to] = true;
                                que.push(e.to);
                            }
                        }
                    }
                    return visited;
                }

            private:
                int _n;
                struct _edge {
                    int to, rev;
                    Cap cap;
                };
                std::vector<std::pair<int, int>> pos;
                std::vector<std::vector<_edge>> g;
        };

}  // namespace atcoder

#include <algorithm>
#include <cassert>
#include <limits>
#include <queue>
#include <vector>

namespace atcoder {

    template <class Cap, class Cost>
        struct mcf_graph {
            public:
                mcf_graph() {}
                mcf_graph(int n) : _n(n), g(n) {}

                int add_edge(int from, int to, Cap cap, Cost cost) {
                    assert(0 <= from && from < _n);
                    assert(0 <= to && to < _n);
                    int m = int(pos.size());
                    pos.push_back({from, int(g[from].size())});
                    g[from].push_back(_edge{to, int(g[to].size()), cap, cost});
                    g[to].push_back(_edge{from, int(g[from].size()) - 1, 0, -cost});
                    return m;
                }

                struct edge {
                    int from, to;
                    Cap cap, flow;
                    Cost cost;
                };

                edge get_edge(int i) {
                    int m = int(pos.size());
                    assert(0 <= i && i < m);
                    auto _e = g[pos[i].first][pos[i].second];
                    auto _re = g[_e.to][_e.rev];
                    return edge{
                        pos[i].first, _e.to, _e.cap + _re.cap, _re.cap, _e.cost,
                    };
                }
                std::vector<edge> edges() {
                    int m = int(pos.size());
                    std::vector<edge> result(m);
                    for (int i = 0; i < m; i++) {
                        result[i] = get_edge(i);
                    }
                    return result;
                }

                std::pair<Cap, Cost> flow(int s, int t) {
                    return flow(s, t, std::numeric_limits<Cap>::max());
                }
                std::pair<Cap, Cost> flow(int s, int t, Cap flow_limit) {
                    return slope(s, t, flow_limit).back();
                }
                std::vector<std::pair<Cap, Cost>> slope(int s, int t) {
                    return slope(s, t, std::numeric_limits<Cap>::max());
                }
                std::vector<std::pair<Cap, Cost>> slope(int s, int t, Cap flow_limit) {
                    assert(0 <= s && s < _n);
                    assert(0 <= t && t < _n);
                    assert(s != t);
                    // variants (C = maxcost):
                    // -(n-1)C <= dual[s] <= dual[i] <= dual[t] = 0
                    // reduced cost (= e.cost + dual[e.from] - dual[e.to]) >= 0 for all edge
                    std::vector<Cost> dual(_n, 0), dist(_n);
                    std::vector<int> pv(_n), pe(_n);
                    std::vector<bool> vis(_n);
                    auto dual_ref = [&]() {
                        std::fill(dist.begin(), dist.end(),
                                std::numeric_limits<Cost>::max());
                        std::fill(pv.begin(), pv.end(), -1);
                        std::fill(pe.begin(), pe.end(), -1);
                        std::fill(vis.begin(), vis.end(), false);
                        struct Q {
                            Cost key;
                            int to;
                            bool operator<(Q r) const { return key > r.key; }
                        };
                        std::priority_queue<Q> que;
                        dist[s] = 0;
                        que.push(Q{0, s});
                        while (!que.empty()) {
                            int v = que.top().to;
                            que.pop();
                            if (vis[v]) continue;
                            vis[v] = true;
                            if (v == t) break;
                            // dist[v] = shortest(s, v) + dual[s] - dual[v]
                            // dist[v] >= 0 (all reduced cost are positive)
                            // dist[v] <= (n-1)C
                            for (int i = 0; i < int(g[v].size()); i++) {
                                auto e = g[v][i];
                                if (vis[e.to] || !e.cap) continue;
                                // |-dual[e.to] + dual[v]| <= (n-1)C
                                // cost <= C - -(n-1)C + 0 = nC
                                Cost cost = e.cost - dual[e.to] + dual[v];
                                if (dist[e.to] - dist[v] > cost) {
                                    dist[e.to] = dist[v] + cost;
                                    pv[e.to] = v;
                                    pe[e.to] = i;
                                    que.push(Q{dist[e.to], e.to});
                                }
                            }
                        }
                        if (!vis[t]) {
                            return false;
                        }

                        for (int v = 0; v < _n; v++) {
                            if (!vis[v]) continue;
                            // dual[v] = dual[v] - dist[t] + dist[v]
                            //         = dual[v] - (shortest(s, t) + dual[s] - dual[t]) +
                            //         (shortest(s, v) + dual[s] - dual[v]) = - shortest(s,
                            //         t) + dual[t] + shortest(s, v) = shortest(s, v) -
                            //         shortest(s, t) >= 0 - (n-1)C
                            dual[v] -= dist[t] - dist[v];
                        }
                        return true;
                    };
                    Cap flow = 0;
                    Cost cost = 0, prev_cost = -1;
                    std::vector<std::pair<Cap, Cost>> result;
                    result.push_back({flow, cost});
                    while (flow < flow_limit) {
                        if (!dual_ref()) break;
                        Cap c = flow_limit - flow;
                        for (int v = t; v != s; v = pv[v]) {
                            c = std::min(c, g[pv[v]][pe[v]].cap);
                        }
                        for (int v = t; v != s; v = pv[v]) {
                            auto& e = g[pv[v]][pe[v]];
                            e.cap -= c;
                            g[v][e.rev].cap += c;
                        }
                        Cost d = -dual[s];
                        flow += c;
                        cost += c * d;
                        if (prev_cost == d) {
                            result.pop_back();
                        }
                        result.push_back({flow, cost});
                        prev_cost = cost;
                    }
                    return result;
                }

            private:
                int _n;

                struct _edge {
                    int to, rev;
                    Cap cap;
                    Cost cost;
                };

                std::vector<std::pair<int, int>> pos;
                std::vector<std::vector<_edge>> g;
        };

}  // namespace atcoder

#include <algorithm>
#include <utility>
#include <vector>

namespace atcoder {
    namespace internal {

        template <class E>
            struct csr {
                std::vector<int> start;
                std::vector<E> elist;
                csr(int n, const std::vector<std::pair<int, E>>& edges)
                    : start(n + 1), elist(edges.size()) {
                        for (auto e : edges) {
                            start[e.first + 1]++;
                        }
                        for (int i = 1; i <= n; i++) {
                            start[i] += start[i - 1];
                        }
                        auto counter = start;
                        for (auto e : edges) {
                            elist[counter[e.first]++] = e.second;
                        }
                    }
            };

        // Reference:
        // R. Tarjan,
        // Depth-First Search and Linear Graph Algorithms
        struct scc_graph {
            public:
                scc_graph(int n) : _n(n) {}

                int num_vertices() { return _n; }

                void add_edge(int from, int to) { edges.push_back({from, {to}}); }

                // @return pair of (# of scc, scc id)
                std::pair<int, std::vector<int>> scc_ids() {
                    auto g = csr<edge>(_n, edges);
                    int now_ord = 0, group_num = 0;
                    std::vector<int> visited, low(_n), ord(_n, -1), ids(_n);
                    visited.reserve(_n);
                    auto dfs = [&](auto self, int v) -> void {
                        low[v] = ord[v] = now_ord++;
                        visited.push_back(v);
                        for (int i = g.start[v]; i < g.start[v + 1]; i++) {
                            auto to = g.elist[i].to;
                            if (ord[to] == -1) {
                                self(self, to);
                                low[v] = std::min(low[v], low[to]);
                            } else {
                                low[v] = std::min(low[v], ord[to]);
                            }
                        }
                        if (low[v] == ord[v]) {
                            while (true) {
                                int u = visited.back();
                                visited.pop_back();
                                ord[u] = _n;
                                ids[u] = group_num;
                                if (u == v) break;
                            }
                            group_num++;
                        }
                    };
                    for (int i = 0; i < _n; i++) {
                        if (ord[i] == -1) dfs(dfs, i);
                    }
                    for (auto& x : ids) {
                        x = group_num - 1 - x;
                    }
                    return {group_num, ids};
                }

                std::vector<std::vector<int>> scc() {
                    auto ids = scc_ids();
                    int group_num = ids.first;
                    std::vector<int> counts(group_num);
                    for (auto x : ids.second) counts[x]++;
                    std::vector<std::vector<int>> groups(ids.first);
                    for (int i = 0; i < group_num; i++) {
                        groups[i].reserve(counts[i]);
                    }
                    for (int i = 0; i < _n; i++) {
                        groups[ids.second[i]].push_back(i);
                    }
                    return groups;
                }

            private:
                int _n;
                struct edge {
                    int to;
                };
                std::vector<std::pair<int, edge>> edges;
        };

    }  // namespace internal

}  // namespace atcoder

#include <cassert>
#include <vector>

namespace atcoder {

    struct scc_graph {
        public:
            scc_graph() : internal(0) {}
            scc_graph(int n) : internal(n) {}

            void add_edge(int from, int to) {
                int n = internal.num_vertices();
                assert(0 <= from && from < n);
                assert(0 <= to && to < n);
                internal.add_edge(from, to);
            }

            std::vector<std::vector<int>> scc() { return internal.scc(); }

        private:
            internal::scc_graph internal;
    };

}  // namespace atcoder

#include <algorithm>
#include <cassert>
#include <vector>

namespace atcoder {

    template <class S, S (*op)(S, S), S (*e)()>
        struct segtree {
            public:
                segtree() : segtree(0) {}
                segtree(int n) : segtree(std::vector<S>(n, e())) {}
                segtree(const std::vector<S>& v) : _n(int(v.size())) {
                    log = internal::ceil_pow2(_n);
                    size = 1 << log;
                    d = std::vector<S>(2 * size, e());
                    for (int i = 0; i < _n; i++) d[size + i] = v[i];
                    for (int i = size - 1; i >= 1; i--) {
                        update(i);
                    }
                }

                void set(int p, S x) {
                    assert(0 <= p && p < _n);
                    p += size;
                    d[p] = x;
                    for (int i = 1; i <= log; i++) update(p >> i);
                }

                S get(int p) {
                    assert(0 <= p && p < _n);
                    return d[p + size];
                }

                S prod(int l, int r) {
                    assert(0 <= l && l <= r && r <= _n);
                    S sml = e(), smr = e();
                    l += size;
                    r += size;

                    while (l < r) {
                        if (l & 1) sml = op(sml, d[l++]);
                        if (r & 1) smr = op(d[--r], smr);
                        l >>= 1;
                        r >>= 1;
                    }
                    return op(sml, smr);
                }

                S all_prod() { return d[1]; }

                template <bool (*f)(S)>
                    int max_right(int l) {
                        return max_right(l, [](S x) { return f(x); });
                    }
                template <class F>
                    int max_right(int l, F f) {
                        assert(0 <= l && l <= _n);
                        assert(f(e()));
                        if (l == _n) return _n;
                        l += size;
                        S sm = e();
                        do {
                            while (l % 2 == 0) l >>= 1;
                            if (!f(op(sm, d[l]))) {
                                while (l < size) {
                                    l = (2 * l);
                                    if (f(op(sm, d[l]))) {
                                        sm = op(sm, d[l]);
                                        l++;
                                    }
                                }
                                return l - size;
                            }
                            sm = op(sm, d[l]);
                            l++;
                        } while ((l & -l) != l);
                        return _n;
                    }

                template <bool (*f)(S)>
                    int min_left(int r) {
                        return min_left(r, [](S x) { return f(x); });
                    }
                template <class F>
                    int min_left(int r, F f) {
                        assert(0 <= r && r <= _n);
                        assert(f(e()));
                        if (r == 0) return 0;
                        r += size;
                        S sm = e();
                        do {
                            r--;
                            while (r > 1 && (r % 2)) r >>= 1;
                            if (!f(op(d[r], sm))) {
                                while (r < size) {
                                    r = (2 * r + 1);
                                    if (f(op(d[r], sm))) {
                                        sm = op(d[r], sm);
                                        r--;
                                    }
                                }
                                return r + 1 - size;
                            }
                            sm = op(d[r], sm);
                        } while ((r & -r) != r);
                        return 0;
                    }

            private:
                int _n, size, log;
                std::vector<S> d;

                void update(int k) { d[k] = op(d[2 * k], d[2 * k + 1]); }
        };

}  // namespace atcoder

#include <algorithm>
#include <cassert>
#include <numeric>
#include <string>
#include <vector>

namespace atcoder {

    namespace internal {

        std::vector<int> sa_naive(const std::vector<int>& s) {
            int n = int(s.size());
            std::vector<int> sa(n);
            std::iota(sa.begin(), sa.end(), 0);
            std::sort(sa.begin(), sa.end(), [&](int l, int r) {
                    if (l == r) return false;
                    while (l < n && r < n) {
                    if (s[l] != s[r]) return s[l] < s[r];
                    l++;
                    r++;
                    }
                    return l == n;
                    });
            return sa;
        }

        std::vector<int> sa_doubling(const std::vector<int>& s) {
            int n = int(s.size());
            std::vector<int> sa(n), rnk = s, tmp(n);
            std::iota(sa.begin(), sa.end(), 0);
            for (int k = 1; k < n; k *= 2) {
                auto cmp = [&](int x, int y) {
                    if (rnk[x] != rnk[y]) return rnk[x] < rnk[y];
                    int rx = x + k < n ? rnk[x + k] : -1;
                    int ry = y + k < n ? rnk[y + k] : -1;
                    return rx < ry;
                };
                std::sort(sa.begin(), sa.end(), cmp);
                tmp[sa[0]] = 0;
                for (int i = 1; i < n; i++) {
                    tmp[sa[i]] = tmp[sa[i - 1]] + (cmp(sa[i - 1], sa[i]) ? 1 : 0);
                }
                std::swap(tmp, rnk);
            }
            return sa;
        }

        // SA-IS, linear-time suffix array construction
        // Reference:
        // G. Nong, S. Zhang, and W. H. Chan,
        // Two Efficient Algorithms for Linear Time Suffix Array Construction
        template <int THRESHOLD_NAIVE = 10, int THRESHOLD_DOUBLING = 40>
            std::vector<int> sa_is(const std::vector<int>& s, int upper) {
                int n = int(s.size());
                if (n == 0) return {};
                if (n == 1) return {0};
                if (n == 2) {
                    if (s[0] < s[1]) {
                        return {0, 1};
                    } else {
                        return {1, 0};
                    }
                }
                if (n < THRESHOLD_NAIVE) {
                    return sa_naive(s);
                }
                if (n < THRESHOLD_DOUBLING) {
                    return sa_doubling(s);
                }

                std::vector<int> sa(n);
                std::vector<bool> ls(n);
                for (int i = n - 2; i >= 0; i--) {
                    ls[i] = (s[i] == s[i + 1]) ? ls[i + 1] : (s[i] < s[i + 1]);
                }
                std::vector<int> sum_l(upper + 1), sum_s(upper + 1);
                for (int i = 0; i < n; i++) {
                    if (!ls[i]) {
                        sum_s[s[i]]++;
                    } else {
                        sum_l[s[i] + 1]++;
                    }
                }
                for (int i = 0; i <= upper; i++) {
                    sum_s[i] += sum_l[i];
                    if (i < upper) sum_l[i + 1] += sum_s[i];
                }

                auto induce = [&](const std::vector<int>& lms) {
                    std::fill(sa.begin(), sa.end(), -1);
                    std::vector<int> buf(upper + 1);
                    std::copy(sum_s.begin(), sum_s.end(), buf.begin());
                    for (auto d : lms) {
                        if (d == n) continue;
                        sa[buf[s[d]]++] = d;
                    }
                    std::copy(sum_l.begin(), sum_l.end(), buf.begin());
                    sa[buf[s[n - 1]]++] = n - 1;
                    for (int i = 0; i < n; i++) {
                        int v = sa[i];
                        if (v >= 1 && !ls[v - 1]) {
                            sa[buf[s[v - 1]]++] = v - 1;
                        }
                    }
                    std::copy(sum_l.begin(), sum_l.end(), buf.begin());
                    for (int i = n - 1; i >= 0; i--) {
                        int v = sa[i];
                        if (v >= 1 && ls[v - 1]) {
                            sa[--buf[s[v - 1] + 1]] = v - 1;
                        }
                    }
                };

                std::vector<int> lms_map(n + 1, -1);
                int m = 0;
                for (int i = 1; i < n; i++) {
                    if (!ls[i - 1] && ls[i]) {
                        lms_map[i] = m++;
                    }
                }
                std::vector<int> lms;
                lms.reserve(m);
                for (int i = 1; i < n; i++) {
                    if (!ls[i - 1] && ls[i]) {
                        lms.push_back(i);
                    }
                }

                induce(lms);

                if (m) {
                    std::vector<int> sorted_lms;
                    sorted_lms.reserve(m);
                    for (int v : sa) {
                        if (lms_map[v] != -1) sorted_lms.push_back(v);
                    }
                    std::vector<int> rec_s(m);
                    int rec_upper = 0;
                    rec_s[lms_map[sorted_lms[0]]] = 0;
                    for (int i = 1; i < m; i++) {
                        int l = sorted_lms[i - 1], r = sorted_lms[i];
                        int end_l = (lms_map[l] + 1 < m) ? lms[lms_map[l] + 1] : n;
                        int end_r = (lms_map[r] + 1 < m) ? lms[lms_map[r] + 1] : n;
                        bool same = true;
                        if (end_l - l != end_r - r) {
                            same = false;
                        } else {
                            while (l < end_l) {
                                if (s[l] != s[r]) {
                                    break;
                                }
                                l++;
                                r++;
                            }
                            if (l == n || s[l] != s[r]) same = false;
                        }
                        if (!same) rec_upper++;
                        rec_s[lms_map[sorted_lms[i]]] = rec_upper;
                    }

                    auto rec_sa =
                        sa_is<THRESHOLD_NAIVE, THRESHOLD_DOUBLING>(rec_s, rec_upper);

                    for (int i = 0; i < m; i++) {
                        sorted_lms[i] = lms[rec_sa[i]];
                    }
                    induce(sorted_lms);
                }
                return sa;
            }

    }  // namespace internal

    std::vector<int> suffix_array(const std::vector<int>& s, int upper) {
        assert(0 <= upper);
        for (int d : s) {
            assert(0 <= d && d <= upper);
        }
        auto sa = internal::sa_is(s, upper);
        return sa;
    }

    template <class T>
        std::vector<int> suffix_array(const std::vector<T>& s) {
            int n = int(s.size());
            std::vector<int> idx(n);
            iota(idx.begin(), idx.end(), 0);
            sort(idx.begin(), idx.end(), [&](int l, int r) { return s[l] < s[r]; });
            std::vector<int> s2(n);
            int now = 0;
            for (int i = 0; i < n; i++) {
                if (i && s[idx[i - 1]] != s[idx[i]]) now++;
                s2[idx[i]] = now;
            }
            return internal::sa_is(s2, now);
        }

    std::vector<int> suffix_array(const std::string& s) {
        int n = int(s.size());
        std::vector<int> s2(n);
        for (int i = 0; i < n; i++) {
            s2[i] = s[i];
        }
        return internal::sa_is(s2, 255);
    }

    // Reference:
    // T. Kasai, G. Lee, H. Arimura, S. Arikawa, and K. Park,
    // Linear-Time Longest-Common-Prefix Computation in Suffix Arrays and Its
    // Applications
    template <class T>
        std::vector<int> lcp_array(const std::vector<T>& s,
                const std::vector<int>& sa) {
            int n = int(s.size());
            assert(n >= 1);
            std::vector<int> rnk(n);
            for (int i = 0; i < n; i++) {
                rnk[sa[i]] = i;
            }
            std::vector<int> lcp(n - 1);
            int h = 0;
            for (int i = 0; i < n; i++) {
                if (h > 0) h--;
                if (rnk[i] == 0) continue;
                int j = sa[rnk[i] - 1];
                for (; j + h < n && i + h < n; h++) {
                    if (s[j + h] != s[i + h]) break;
                }
                lcp[rnk[i] - 1] = h;
            }
            return lcp;
        }

    std::vector<int> lcp_array(const std::string& s, const std::vector<int>& sa) {
        int n = int(s.size());
        std::vector<int> s2(n);
        for (int i = 0; i < n; i++) {
            s2[i] = s[i];
        }
        return lcp_array(s2, sa);
    }

    // Reference:
    // D. Gusfield,
    // Algorithms on Strings, Trees, and Sequences: Computer Science and
    // Computational Biology
    template <class T>
        std::vector<int> z_algorithm(const std::vector<T>& s) {
            int n = int(s.size());
            if (n == 0) return {};
            std::vector<int> z(n);
            z[0] = 0;
            for (int i = 1, j = 0; i < n; i++) {
                int& k = z[i];
                k = (j + z[j] <= i) ? 0 : std::min(j + z[j] - i, z[i - j]);
                while (i + k < n && s[k] == s[i + k]) k++;
                if (j + z[j] < i + z[i]) j = i;
            }
            z[0] = n;
            return z;
        }

    std::vector<int> z_algorithm(const std::string& s) {
        int n = int(s.size());
        std::vector<int> s2(n);
        for (int i = 0; i < n; i++) {
            s2[i] = s[i];
        }
        return z_algorithm(s2);
    }

}  // namespace atcoder

#include <cassert>
#include <vector>

namespace atcoder {

    // Reference:
    // B. Aspvall, M. Plass, and R. Tarjan,
    // A Linear-Time Algorithm for Testing the Truth of Certain Quantified Boolean
    // Formulas
    struct two_sat {
        public:
            two_sat() : _n(0), scc(0) {}
            two_sat(int n) : _n(n), _answer(n), scc(2 * n) {}

            void add_clause(int i, bool f, int j, bool g) {
                assert(0 <= i && i < _n);
                assert(0 <= j && j < _n);
                scc.add_edge(2 * i + (f ? 0 : 1), 2 * j + (g ? 1 : 0));
                scc.add_edge(2 * j + (g ? 0 : 1), 2 * i + (f ? 1 : 0));
            }
            bool satisfiable() {
                auto id = scc.scc_ids().second;
                for (int i = 0; i < _n; i++) {
                    if (id[2 * i] == id[2 * i + 1]) return false;
                    _answer[i] = id[2 * i] < id[2 * i + 1];
                }
                return true;
            }
            std::vector<bool> answer() { return _answer; }

        private:
            int _n;
            std::vector<bool> _answer;
            internal::scc_graph scc;
    };

}  // namespace atcoder

using namespace std;
int main() {}