#include <bits/stdc++.h>
#ifndef ATCODER_INTERNAL_BITOP_HPP
#define ATCODER_INTERNAL_BITOP_HPP 1
#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
#endif // ATCODER_INTERNAL_BITOP_HPP
#ifndef ATCODER_INTERNAL_MATH_HPP
#define ATCODER_INTERNAL_MATH_HPP 1
#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
#endif // ATCODER_INTERNAL_MATH_HPP
#ifndef ATCODER_INTERNAL_QUEUE_HPP
#define ATCODER_INTERNAL_QUEUE_HPP 1
#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
#endif // ATCODER_INTERNAL_QUEUE_HPP
#ifndef ATCODER_INTERNAL_SCC_HPP
#define ATCODER_INTERNAL_SCC_HPP 1
#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
#endif // ATCODER_INTERNAL_SCC_HPP
#ifndef ATCODER_INTERNAL_TYPE_TRAITS_HPP
#define ATCODER_INTERNAL_TYPE_TRAITS_HPP 1
#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
#endif // ATCODER_INTERNAL_TYPE_TRAITS_HPP
#ifndef ATCODER_MODINT_HPP
#define ATCODER_MODINT_HPP 1
#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
#endif // ATCODER_MODINT_HPP
#ifndef ATCODER_CONVOLUTION_HPP
#define ATCODER_CONVOLUTION_HPP 1
#include <algorithm>
#include <array>
#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
#endif // ATCODER_CONVOLUTION_HPP
#ifndef ATCODER_DSU_HPP
#define ATCODER_DSU_HPP 1
#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
#endif // ATCODER_DSU_HPP
#ifndef ATCODER_FENWICKTREE_HPP
#define ATCODER_FENWICKTREE_HPP 1
#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
#endif // ATCODER_FENWICKTREE_HPP
#ifndef ATCODER_LAZYSEGTREE_HPP
#define ATCODER_LAZYSEGTREE_HPP 1
#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
#endif // ATCODER_LAZYSEGTREE_HPP
#ifndef ATCODER_MATH_HPP
#define ATCODER_MATH_HPP 1
#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
#endif // ATCODER_MATH_HPP
#ifndef ATCODER_MAXFLOW_HPP
#define ATCODER_MAXFLOW_HPP 1
#include <algorithm>
#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
#endif // ATCODER_MAXFLOW_HPP
#ifndef ATCODER_MINCOSTFLOW_HPP
#define ATCODER_MINCOSTFLOW_HPP 1
#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
#endif // ATCODER_MINCOSTFLOW_HPP
#ifndef ATCODER_SCC_HPP
#define ATCODER_SCC_HPP 1
#include <algorithm>
#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
#endif // ATCODER_SCC_HPP
#ifndef ATCODER_SEGTREE_HPP
#define ATCODER_SEGTREE_HPP 1
#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
#endif // ATCODER_SEGTREE_HPP
#ifndef ATCODER_STRING_HPP
#define ATCODER_STRING_HPP 1
#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
#endif // ATCODER_STRING_HPP
#ifndef ATCODER_TWOSAT_HPP
#define ATCODER_TWOSAT_HPP 1
#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
#endif // ATCODER_TWOSAT_HPP
#pragma GCC optimize("Ofast")
#pragma GCC optimize("O3")
#pragma GCC optimize("unroll-loops")
#pragma GCC target("avx")
#include <bits/stdc++.h>
#define _GLIBCXX_DEBUG
using namespace std;
using namespace atcoder;
using ll = long long;
using vec = vector<ll>;
using vect = vector<double>;
using Graph = vector<vector<ll>>;
#define endl '\n'
#define loop(i, n) for (register int i = 0; i < n; i++)
#define Loop(i, m, n) for (register int i = m; i < n; i++)
#define pool(i, n) for (register int i = n; i >= 0; i--)
#define Pool(i, m, n) for (register int i = n; i >= m; i--)
#define modd 1000000007ll
//#define modd 998244353ll
#define flagcount(bit) __builtin_popcount(bit)
#define flag(x) (1ll << x)
#define flagadd(bit, x) bit |= flag(x)
#define flagpop(bit, x) bit &= ~flag(x)
#define flagon(bit, i) bit &flag(i)
#define flagoff(bit, i) !(bit & (1ll << i))
#define all(v) v.begin(), v.end()
#define low2way(v, x) lower_bound(all(v), x)
#define high2way(v, x) upper_bound(all(v), x)
#define idx_lower(v, x) (distance(v.begin(), low2way(v, x)))  //配列vでx未満の要素数を返す
#define idx_upper(v, x) (distance(v.begin(), high2way(v, x))) //配列vでx以下の要素数を返す
#define idx_lower2(v, x) (v.size() - idx_lower(v, x))         //配列vでx以上の要素数を返す
#define idx_upper2(v, x) (v.size() - idx_upper(v, x))         //配列vでxより大きい要素の数を返す
#define putout(a) cout << a << '\n'
#define Sum(v) accumulate(all(v), 0ll)
ll ctoi(char c)
{
    if (c >= '0' && c <= '9')
    {
        return c - '0';
    }
    return -1;
}
template <typename T>
string make_string(T N)
{
    string ret;
    T now = N;
    while (now > 0)
    {
        T x = now % 10;
        ret += (char)('0' + x);
        now /= 10;
    }
    reverse(all(ret));
    return ret;
}
template <typename T>
T gcd(T a, T b)
{
    if (a % b == 0)
    {
        return (b);
    }
    else
    {
        return (gcd(b, a % b));
    }
}
template <typename T>
T lcm(T x, T y)
{
    T z = gcd(x, y);
    return x * y / z;
}
template <typename T>
bool primejudge(T n)
{
    if (n < 2)
        return false;
    else if (n == 2)
        return true;
    else if (n % 2 == 0)
        return false;
    double sqrtn = sqrt(n);
    for (T i = 3; i < sqrtn + 1; i++)
    {
        if (n % i == 0)
        {
            return false;
        }
        i++;
    }
    return true;
}
template <typename T>
bool chmax(T &a, const T &b)
{
    if (a < b)
    {
        a = b; // aをbで更新
        return true;
    }
    return false;
}
template <typename T>
bool chmin(T &a, const T &b)
{
    if (a > b)
    {
        a = b; // aをbで更新
        return true;
    }
    return false;
}
//場合によって使い分ける
//const ll dx[4]={1,0,-1,0};
//const ll dy[4]={0,1,0,-1};
const ll dx[8] = {1, 1, 0, -1, -1, -1, 0, 1};
const ll dy[8] = {0, 1, 1, 1, 0, -1, -1, -1};

//vector<vector<ll>> field(h, vector<ll>(w));
static const double pi = acos(-1.0);
//誤差チェッカー
#define EPS (1e-10)
#define equals(a, b) (fabs((a) - (b)) < EPS)
//点を表わす構造体
struct Point
{
public:
    double x, y;
    Point(double _x = 0, double _y = 0) : x(_x), y(_y) {}
    /*
    以下ベクトルに対する演算子の定義
    +:ベクトルの足し算
    -:ベクトルの引き算
    *:スカラー倍
    /:スカラーの逆数倍
    */
    Point operator+(const Point p) { return Point(x + p.x, y + p.y); }
    Point operator-(const Point p) { return Point(x - p.x, y - p.y); }
    Point operator*(const double k) { return Point(x * k, y * k); }
    Point operator/(const double k) { return Point(x / k, y / k); }
    double abs() { return sqrt(norm()); }
    double norm() { return x * x + y * y; }
    //ベクトルの比較(x軸方向のベクトルから比較)
    bool operator<(const Point &p) const
    {
        return x != p.x ? x < p.x : y < p.y;
    }
    bool operator>(const Point &p) const
    {
        return x != p.x ? x > p.x : y > p.y;
    }
    bool operator==(const Point &p) const
    {
        return equals(x, p.x) && equals(y, p.y);
    }
    bool operator!=(const Point &p) const
    {
        return (!equals(x, p.x)) || (!equals(y, p.y));
    }
};
//ベクトルとして扱う場合
typedef Point Vector;
//線分を表わす構造体
struct Segment
{
    Point p1, p2;
    Segment() = default;
    Segment(Point _p1, Point _p2) : p1(_p1), p2(_p2) {}
};
//直線として扱う場合
typedef Segment Line;
//ベクトルのノルム
double norm(Vector a)
{
    return a.x * a.x + a.y * a.y;
}
//ベクトルの大きさ
double abs(Vector a)
{
    return sqrt(norm(a));
}
//ベクトルの内積
double dot(Vector a, Vector b)
{
    return a.x * b.x + a.y * b.y;
}
//ベクトルの外積
double cross(Vector a, Vector b)
{
    return a.x * b.y - a.y * b.x;
}
//内積≓0かどうかによるベクトルの直交判定(様々な引数で答えられるように複数用意)
bool Orthogonal(Vector a, Vector b)
{
    return equals(dot(a, b), 0.0);
}
bool Orthogonal(Point a1, Point a2, Point b1, Point b2)
{
    return Orthogonal(a1 - a2, b1 - b2);
}
bool Orthogonal(Segment s1, Segment s2)
{
    return equals(dot(s1.p2 - s1.p1, s2.p2 - s2.p1), 0.0);
}
//外積≓0かどうかによるベクトルの平行判定(様々な引数で答えられるように複数用意)
bool Parallel(Vector a, Vector b)
{
    return equals(cross(a, b), 0.0);
}
bool Parallel(Point a1, Point a2, Point b1, Point b2)
{
    return Parallel(a1 - a2, b1 - b2);
}
bool Parallel(Segment s1, Segment s2)
{
    return equals(cross(s1.p2 - s1.p1, s2.p2 - s2.p1), 0.0);
}
//直線sに対して点ｐから垂線を引いたときの交点x(直線sに対する点pの射影)を求める
Point Project(Line s, Point p)
{
    Vector base = s.p2 - s.p1, hypo = p - s.p1;
    double scalar = (dot(base, hypo) / norm(base));
    return s.p1 + base * scalar;
}
//直線sに対して点pと線対称の位置にある点p'を求める
Point Reflect(Line s, Point p)
{
    return p + (Project(s, p) - p) * 2.0;
}
//点と点の距離
double GetDist(Point a, Point b)
{
    return abs(a - b);
}
//直線と点の距離
double GetDistLP(Line l, Point p)
{
    return abs(Project(l, p) - p);
}
//線分と点の距離
double GetDistSP(Segment s, Point p)
{
    if (dot(s.p2 - s.p1, p - s.p1) < 0.0)
        return abs(p - s.p1);
    if (dot(s.p1 - s.p2, p - s.p2) < 0.0)
        return abs(p - s.p2);
    return GetDistLP(s, p);
}
//Counter-Clockwise(AOJ:CGL_1_Cより)
static const int COUNTER_CLOCKWISE = 1; //反時計回り
static const int CLOCKWISE = -1;        //時計回り
static const int ONLINE_BACK = 2;       //線分の後ろにある
static const int ONLINE_FRONT = -2;     //線分の前にある
static const int ON_SEGMENT = 0;        //線分上にある
//線分p0p1と点p2の位置関係を求める
int CCW(Point p0, Point p1, Point p2)
{
    Vector a = p1 - p0, b = p2 - p0; //a:p0->p1,b:p0->p2
    if (cross(a, b) > EPS)
        return COUNTER_CLOCKWISE; //外積が正なら反時計回り(sinθ>0)
    if (cross(a, b) < -EPS)
        return CLOCKWISE; //外積が負なら時計回り(sinθ<0)
    if (dot(a, b) < -EPS)
        return ONLINE_BACK; //外積が0かつ内積が負(cosθ<0)
    if (a.norm() < b.norm())
        return ONLINE_FRONT; //p2がp0p1に含まれないときp0p2=p0p1+p1p2となり|a|^2<|b|^2が成立
    return ON_SEGMENT;       //最後にp0p2p1が残る
}
//2本の線分の交差判定(点と線分で2種類用意)
bool Intersect(Point p1, Point p2, Point p3, Point p4)
{
    return (CCW(p1, p2, p3) * CCW(p1, p2, p4) <= 0 && CCW(p3, p4, p1) * CCW(p3, p4, p2) <= 0);
}
bool Intersect(Segment s1, Segment s2)
{
    return Intersect(s1.p1, s1.p2, s2.p1, s2.p2);
}
//線分と線分の距離
double GetDist(Segment s1, Segment s2)
{
    //線分が交差していた場合、距離は0.0
    if (Intersect(s1, s2))
        return 0.0;
    return min(min(GetDistSP(s1, s2.p1), GetDistSP(s1, s2.p2)), min(GetDistSP(s2, s1.p1), GetDistSP(s2, s1.p2)));
}
//交差する2本の線分の交点
Point CrossPoint(Segment s1, Segment s2)
{
    if (!Intersect(s1, s2))
        cout << "2本の線分は交点を持たない" << endl; //デバッグ用
    Vector base = s2.p2 - s2.p1, hypo1 = s1.p1 - s2.p1, hypo2 = s1.p2 - s2.p1;
    double d1 = fabs(cross(base, hypo1)) / abs(base);
    double d2 = fabs(cross(base, hypo2)) / abs(base);
    Point x = s1.p1 + (s1.p2 - s1.p1) * (d1 / (d1 + d2));
    return x;
}
//座標cを中心とする半径rの円を表わす構造体
struct Circle
{
public:
    Point c;
    double r;
    Circle(Point _c, double _r) : c(_c), r(_r) {}
};
//円と直線の交点の組(接点なら同じものを2つ)を求める(小さい順に入れる)
pair<Point, Point> CrossPoints(Circle c, Line l)
{
    if (GetDistLP(l, c.c) - c.r > EPS)
        cout << "円と直線は交点を持たない" << endl;      //デバッグ用
    Point mid = Project(l, c.c);                         //交点の中間
    Vector e = (l.p2 - l.p1) / abs(l.p2 - l.p1);         //直線lの単位ベクトル
    double halfdist = sqrt(c.r * c.r - norm(c.c - mid)); //三平方の定理で2交点の距離の半分を求める
    Point s = mid + e * halfdist, t = mid - e * halfdist;
    if (s > t)
        swap(s, t);
    return make_pair(s, t); //s<tの順
}
//ベクトルの角度
double arg(Vector p)
{
    return atan2(p.y, p.x);
}
//極座標から直交座標に変換
Vector polar(double r, double theta)
{
    return Point(cos(theta) * r, sin(theta) * r);
}
//円と円の交点の組(接点なら同じものを2つ)を求める(小さい順に入れる)
pair<Point, Point> CrossPoints(Circle c1, Circle c2)
{
    if (GetDist(c1.c, c2.c) - (c1.r + c2.r) > EPS)
        cout << "2円は交点を持たない" << endl;                             //デバッグ用
    double d = abs(c2.c - c1.c);                                           //中心間距離
    double a = acos((d * d + c1.r * c1.r - c2.r * c2.r) / (2 * d * c1.r)); //余弦定理+arccos
    double t = arg(c2.c - c1.c);                                           //偏角
    Point p1 = c1.c + polar(c1.r, t + a), p2 = c1.c + polar(c1.r, t - a);
    if (p1 > p2)
        swap(p1, p2);
    return make_pair(p1, p2);
}
//多角形は点の列として扱う
typedef vector<Point> Polygon;
//点の内包判定
/*
    IN 2
    ON 1
    OUT 0
*/
int Contain(Polygon g, Point p)
{
    int n = g.size();
    bool x = false;
    for (int i = 0; i < n; i++)
    {
        Vector a = g[i] - p;           //p->g[i]
        Vector b = g[(i + 1) % n] - p; //p->g[i+1] 但しn-1の次は0
        if (abs(cross(a, b)) < EPS && dot(a, b) < EPS)
            return 1; //外積の大きさが0で内積0以下なら線分上
        if (a.y > b.y)
            swap(a, b);
        if (a.y < EPS && EPS < b.y && cross(a, b) > EPS)
            x = (!x); //xの真偽値を反転
    }
    return (x ? 2 : 0); //xが真なら2(内包)を、偽なら0(外)を返す
}
//凸包を求める
Polygon AndrewScan(Polygon s)
{
    Polygon u, l;
    if (s.size() < 3)
        return s;
    sort(s.begin(), s.end()); //x座標(y座標)の順でソート
    //xが小さい順に2つ追加
    u.push_back(s[0]);
    u.push_back(s[1]);
    //xが大きい順に2つ追加
    l.push_back(s[s.size() - 1]);
    l.push_back(s[s.size() - 2]);
    //凸包の上部生成
    for (int i = 2; i < s.size(); i++)
    {
        for (int n = u.size(); n >= 2 && CCW(u[n - 2], u[n - 1], s[i]) == COUNTER_CLOCKWISE; n--)
        {
            u.pop_back(); //凸にならなかったら頂点を削除
        }
        u.push_back(s[i]);
    }
    //凸包の下部生成
    for (int i = s.size() - 3; i >= 0; i--)
    {
        for (int n = l.size(); n >= 2 && CCW(l[n - 2], l[n - 1], s[i]) == COUNTER_CLOCKWISE; n--)
        {
            l.pop_back();
        }
        l.push_back(s[i]);
    }
    //反時計回りになるように凸包の点列を生成
    reverse(l.begin(), l.end());
    for (int i = u.size() - 2; i >= 1; i--)
        l.push_back(u[i]);
    return l;
}
#define P complex<double>
#define ld double
ld cross(const P &l, const P &r)
{
    return l.real() * r.imag() - l.imag() * r.real();
}
template <class iter>
std::pair<P, ld> min_ball(iter left, iter right, int seed = 1333)
{
    const int n = right - left;

    assert(n >= 1);
    if (n == 1)
    {
        return {*left, ld(0)};
    }

    std::mt19937 mt(seed);
    std::shuffle(left, right, mt);
    // std::random_shuffle(left, right); // simple but deprecated

    iter ps = left;
    using circle = std::pair<P, ld>;

    auto make_circle_3 = [](const P &a, const P &b, const P &c) -> circle {
        ld A = std::norm(b - c), B = std::norm(c - a), C = std::norm(a - b),
           S = cross(b - a, c - a);
        P p = (A * (B + C - A) * a + B * (C + A - B) * b + C * (A + B - C) * c) / (4 * S * S);
        ld r2 = std::norm(p - a);
        return {p, r2};
    };

    auto make_circle_2 = [](const P &a, const P &b) -> circle {
        P c = (a + b) / (ld)2;
        ld r2 = std::norm(a - c);
        return {c, r2};
    };

    auto in_circle = [](const P &a, const circle &c) -> bool {
        return std::norm(a - c.first) <= c.second + EPS;
    };

    circle c = make_circle_2(ps[0], ps[1]);

    // MiniDisc
    for (int i = 2; i < n; ++i)
    {
        if (!in_circle(ps[i], c))
        {
            // MiniDiscWithPoint
            c = make_circle_2(ps[0], ps[i]);
            for (int j = 1; j < i; ++j)
            {
                if (!in_circle(ps[j], c))
                {
                    // MiniDiscWith2Points
                    c = make_circle_2(ps[i], ps[j]);
                    for (int k = 0; k < j; ++k)
                    {
                        if (!in_circle(ps[k], c))
                        {
                            c = make_circle_3(ps[i], ps[j], ps[k]);
                        }
                    }
                }
            }
        }
    }
    return c;
}
#undef ld
int main()
{
    cout << fixed << setprecision(15);
    ll N;
    cin >> N;
    vector<P> ps(N);
    loop(i, N)
    {
        double x, y;
        cin >> x >> y;
        ps[i].real(x);
        ps[i].imag(y);
    }
    double ans = sqrt(min_ball(all(ps)).second);
    putout(ans);
    return 0;
}


Measured memory_bytes: 3872