結果

問題 No.8046 yukicoderの過去問
ユーザー ttttan2ttttan2
提出日時 2022-06-08 23:55:23
言語 C++17
(gcc 12.3.0 + boost 1.83.0)
結果
TLE  
実行時間 -
コード長 56,774 bytes
コンパイル時間 4,101 ms
コンパイル使用メモリ 256,960 KB
実行使用メモリ 42,316 KB
最終ジャッジ日時 2024-09-21 05:17:23
合計ジャッジ時間 11,394 ms
ジャッジサーバーID
(参考情報)
judge2 / judge3
このコードへのチャレンジ
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テストケース

テストケース表示
入力 結果 実行時間
実行使用メモリ
testcase_00 AC 890 ms
40,724 KB
testcase_01 AC 1,198 ms
35,344 KB
testcase_02 AC 1,157 ms
35,340 KB
testcase_03 TLE -
testcase_04 -- -
testcase_05 -- -
testcase_06 -- -
testcase_07 -- -
testcase_08 -- -
権限があれば一括ダウンロードができます

ソースコード

diff #

#include <algorithm>
#include <bitset>
#include <complex>
#include <deque>
#include <exception>
#include <fstream>
#include <functional>
#include <iomanip>
#include <ios>
#include <iosfwd>
#include <iostream>
#include <istream>
#include <iterator>
#include <limits>
#include <list>
#include <locale>
#include <map>
#include <memory>
#include <new>
#include <numeric>
#include <ostream>
#include <queue>
#include <set>
#include <sstream>
#include <stack>
#include <stdexcept>
#include <streambuf>
#include <string>
#include <typeinfo>
#include <utility>
#include <valarray>
#include <vector>
#if __cplusplus >= 201103L
#include <array>
#include <atomic>
#include <chrono>
#include <condition_variable>
#include <forward_list>
#include <future>
#include <initializer_list>
#include <mutex>
#include <random>
#include <ratio>
#include <regex>
#include <scoped_allocator>
#include <system_error>
#include <thread>
#include <tuple>
#include <typeindex>
#include <type_traits>
#include <unordered_map>
#include <unordered_set>
#endif
using namespace std;
typedef long long ll;
typedef unsigned long long ull;
typedef unsigned int ui;
typedef pair<int,int> pii;
typedef pair<pii,int> ppii;
typedef pair<int,pii> pipi;
typedef pair<ll,ll> pll;
typedef pair<pll,ll> ppll;
typedef pair<ll,pll> plpl;
typedef pair<pii,pii> pippi;
typedef tuple<ll,ll,ll> tl;
typedef pair<long double,long double> pdd;
//typedef vector<vector<ll>> mat;
const ll mod=1000000007;
const ll mod2=998244353;
const ll mod3=1000000009;
ll inf=numeric_limits<ll>::max()/2;
int iinf=numeric_limits<int>::max()/2;
double pi=3.14159265358979323846;
double eps=1e-10;
#define rep(i,m,n) for(ll i=m;i<n;i++)
#define rrep(i,n,m) for(ll i=n;i>=m;i--)
#define srep(itr,st) for(auto itr=st.begin();itr!=st.end();itr++)
#define mrep(itr,mp) for(auto& itr:mp)
#define Max(a,b) a=max(a,b)
#define Min(a,b) a=min(a,b)
int dh[4]={1,0,-1,0};
int dw[4]={0,1,0,-1};
int ddh[8]={-1,-1,-1,0,0,1,1,1};
int ddw[8]={-1,0,1,-1,1,-1,0,1};
struct custom_hash {
    static uint64_t splitmix64(uint64_t x) {
        x+=0x9e3779b97f4a7c15;x=(x^(x>>30))*0xbf58476d1ce4e5b9;x=(x^(x>>27))*0x94d049bb133111eb;
        return x^(x>>31);}
    size_t operator()(uint64_t x) const {
        static const uint64_t FIXED_RANDOM = chrono::steady_clock::now().time_since_epoch().count();
        return splitmix64(x + FIXED_RANDOM);}
};
#define umh unordered_map<int,ll,custom_hash>
ll gcd(ll a,ll b){
    if(a<0)a=-a;if(b<0)b=-b;if(a<b)swap(a,b);if(b==0)return a;if(a%b==0)return b;return gcd(b,a%b);}
ll lcm(ll a,ll b){if(a==0)return b;if(b==0)return a;ll c=gcd(a,b);a/=c;a*=b;return a;}
ll Pow(ll n,ll k){
    if(k<0)return 0;ll ret=1;ll now=n;while(k>0){if(k&1)ret*=now;now*=now;k/=2;}return ret;
}
ll beki(ll n,ll k,ll md){
  ll ret=1;ll now=n;now%=md;
  while(k>0){
    if(k%2==1){ret*=now;ret%=md;}
    now*=now;now%=md;k=k>>1;}return ret;
}
ll gyaku(ll n,ll md){return beki(n,md-2,md);}
ll popcount(ll n){ll ret=0;ll u=n;while(u>0){ret+=u%2;u/=2;}return ret;}
#ifndef ATCODER_INTERNAL_BITOP_HPP
#define ATCODER_INTERNAL_BITOP_HPP 1
#ifdef _MSC_VER
#include <intrin.h>
#endif
namespace atcoder {
namespace internal {
int ceil_pow2(int n) {
    int x = 0;while ((1U << x) < (unsigned int)(n)) x++;return x;}
int bsf(unsigned int n) {
#ifdef _MSC_VER
    unsigned long index;
    _BitScanForward(&index, n);
    return index;
#else
    return __builtin_ctz(n);
#endif
}}}
#endif
#ifndef ATCODER_INTERNAL_MATH_HPP
#define ATCODER_INTERNAL_MATH_HPP 1
#include <utility>
namespace atcoder {
namespace internal {
constexpr long long safe_mod(long long x, long long m) {
    x%=m;if(x<0)x+=m;return x;}
struct barrett {
    unsigned int _m;
    unsigned long long im;
    barrett(unsigned int m) : _m(m), im((unsigned long long)(-1) / m + 1) {}
    unsigned int umod() const { return _m; }
    unsigned int mul(unsigned int a, unsigned int b) const {
        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;
    }
};
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;
}

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);
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};
    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;
        auto tmp = s;
        s = t;
        t = tmp;
        tmp = m0;
        m0 = m1;
        m1 = tmp;
    }
    if (m0 < 0) m0 += b / s;
    return {s, m0};
}
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);
}
}
#endif
#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++; }
};
}
}
#endif
#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;}
    }
};
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}}); }
    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;
};
}
}
#endif
#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;
}
}
#endif
#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>;
}
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>;
}
}
#endif
#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];
    if (first) {
        first = false;
        mint es[30], ies[30];
        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--) {
            
            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];
    if (first) {
        first = false;
        mint es[30], ies[30];
        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--) {
            
            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))];
        }
    }
}
}
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;
    static constexpr unsigned long long MOD2 = 167772161;
    static constexpr unsigned long long MOD3 = 469762049;
    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;
        
        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;
}
}
#endif
#ifndef ATCODER_DSU_HPP
#define ATCODER_DSU_HPP 1
#include <algorithm>
#include <cassert>
#include <vector>
namespace atcoder {

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;
    
    std::vector<int> parent_or_size;
};
}
#endif
#ifndef ATCODER_FENWICKTREE_HPP
#define ATCODER_FENWICKTREE_HPP 1
#include <cassert>
#include <vector>
namespace atcoder {

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;
        }
    }
    void change(int p,T x){
        add(p,x-data[p]);
    }
    T sum(int l, int r) {
        assert(0 <= l && l <= r && r <= _n);
        return sum(r) - sum(l);
    }
    int lb(T w){
        if(w<=0)return 0;
        int x=0;
        int u=1;while(u<_n)u*=2;
        for(int k=u;k>0;k/=2){
            if(x+k<=_n&&data[x+k-1]<w){
                w-=data[x+k-1];
                x+=k;
            }
        }
        return x;
    }
  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;
    }
};
}
#endif
#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();
    }
};
}
#endif
#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;
};
}
#endif
#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);
       
        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;
                
                for (int i = 0; i < int(g[v].size()); i++) {
                    auto e = g[v][i];
                    if (vis[e.to] || !e.cap) continue;
                    
                    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] -= 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;
};
}
#endif
#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]); }
};
}
#endif

using namespace atcoder;
#define endl "\n"
//using mint=static_modint<mod2>;
//mintに注意
//除算はa[0]!=0でないとできない
template < int MOD >
struct mint {
public:
    long long x;
    mint(long long x = 0) : x((x % MOD + MOD) % MOD) {}
    mint(std::string &s) {
        long long z = 0;
        for (int i = 0; i < s.size(); i++) {
            z *= 10;
            z += s[i] - '0';
            z %= MOD;
        }
        this->x = z;
    }
    mint &operator+=(const mint &a) {
        if ((x += a.x) >= MOD) x -= MOD;
        return *this;
    }
    mint &operator-=(const mint &a) {
        if ((x += MOD - a.x) >= MOD) x -= MOD;
        return *this;
    }
    mint &operator*=(const mint &a) {
        (x *= a.x) %= MOD;
        return *this;
    }
    mint &operator/=(const mint &a) {
        long long n = MOD - 2;
        mint u = 1, b = a;
        while (n > 0) {
            if (n & 1) {
                u *= b;
            }
            b *= b;
            n >>= 1;
        }
        return *this *= u;
    }
    mint operator+(const mint &a) const {
        mint res(*this);
        return res += a;
    }
    mint operator-() const {
        return mint() -= *this;
    }
    mint operator-(const mint &a) const {
        mint res(*this);
        return res -= a;
    }
    mint operator*(const mint &a) const {
        mint res(*this);
        return res *= a;
    }
    mint operator/(const mint &a) const {
        mint res(*this);
        return res /= a;
    }
    friend std::ostream &operator<<(std::ostream &os, const mint &n) {
        return os << n.x;
    }
    friend std::istream &operator>>(std::istream &is, mint &n) {
        long long x;
        is >> x;
        n = mint(x);
        return is;
    }
    bool operator==(const mint &a) const {
        return this->x == a.x;
    }
    bool operator!=(const mint &a) const {
        return this->x != a.x;
    }
    mint pow(long long k) const {
        mint ret = 1;
        mint p = this->x;
        while (k > 0) {
            if (k & 1) {
                ret *= p;
            }
            p *= p;
            k >>= 1;
        }
        return ret;
    }
};
//constexpr int MOD = mod;
 
//any=false:NTT friendly  any=true:not NTT friendly
template < const int MOD, bool any = false >
struct FormalPowerSeries {
private:
    using P = FormalPowerSeries< MOD, any >;
    template < class T, class F = multiplies< T > >
    T power(T a, long long n, F op = multiplies< T >(), T e = {1}) const {
        assert(n >= 0);
        T res = e;
        while (n) {
            if (n & 1) res = op(res, a);
            if (n >>= 1) a = op(a, a);
        }
        return res;
    }
    template < int _MOD >
    void ntt(vector< mint< _MOD > > &a, bool inverse) {
        static vector< mint< _MOD > > dw(30), idw(30);
        if (dw[0] == 0) {
            mint< _MOD > root = 2;
            while (power(root, (_MOD - 1) / 2) == 1) root += 1;
            for (int i = 0; i < 30; i++) dw[i] = -power(root, (_MOD - 1) >> (i + 2)), idw[i] = mint< _MOD >(1) / dw[i];
        }
        int n = a.size();
        assert((n & (n - 1)) == 0);
        if (not inverse) {
            for (int m = n; m >>= 1;) {
                mint< _MOD > w = 1;
                for (int s = 0, k = 0; s < n; s += 2 * m) {
                    for (int i = s, j = s + m; i < s + m; i++, j++) {
                        auto x = a[i], y = a[j] * w;
                        if (x.x >= _MOD) x.x -= _MOD;
                        a[i].x = x.x + y.x, a[j].x = x.x + (_MOD - y.x);
                    }
                    w *= dw[__builtin_ctz(++k)];
                }
            }
        } else {
            for (int m = 1; m < n; m *= 2) {
                mint< _MOD > w = 1;
                for (int s = 0, k = 0; s < n; s += 2 * m) {
                    for (int i = s, j = s + m; i < s + m; i++, j++) {
                        auto x = a[i], y = a[j];
                        a[i] = x + y, a[j].x = x.x + (_MOD - y.x), a[j] *= w;
                    }
                    w *= idw[__builtin_ctz(++k)];
                }
            }
        }
        auto c = mint< _MOD >(1) / mint< _MOD >(inverse ? n : 1);
        for (auto &&e : a) e *= c;
    }
    template < int _MOD >
    vector< mint< _MOD > > convolution(vector< mint< _MOD > > l, vector< mint< _MOD > > r) {
        if (l.empty() || r.empty()) return {};
        int n = l.size(), m = r.size(), sz = 1 ;
        while(sz<=2*(n+m-1)-1)sz*=2;
        if (min(n, m) < 30) {
            vector< long long > res(n + m - 1);
            for (int i = 0; i < n; i++)
                for (int j = 0; j < m; j++) res[i + j] += (l[i] * r[j]).x;
            return {begin(res), end(res)};
        }
        bool eq = l == r;
        l.resize(sz), ntt(l, false);
        if (eq) r = l;
        else
            r.resize(sz), ntt(r, false);
        for (int i = 0; i < sz; i++) l[i] *= r[i];
        ntt(l, true), l.resize(n + m - 1);
        return l;
    }
    P pre(const P &p, int sz) const {
        P ret;
        ret.a = vector< mint< MOD > >(p.a.begin(), p.a.begin() + min((int)p.a.size(), sz));
        return ret;
    }
 
public:
    vector< mint< MOD > > a;
    FormalPowerSeries(int sz = 0) {
        this->a.resize(sz, 0);
    }
 
    P resize(int k) const {
        return pre(*this, k);
    }
    FormalPowerSeries(std::initializer_list< mint< MOD > > v) {
        this->a = v;
    }
 
    size_t size() const {
        return this->a.size();
    }
    bool operator<(const P &r) const {
        return this->a.size() < r.a.size();
    }
    bool operator>(const P &r) const {
        return this->a.size() > r.a.size();
    }
    P operator+(const P &a) const {
        return P(*this) += a;
    }
    //定数を足す
    P operator+(const long long a) const {
        return P(*this) += a;
    }
    P operator-(const P &a) const {
        return P(*this) -= a;
    }
    P operator*(const P &a) const {
        return P(*this) *= a;
    }
    P operator*(const long long a) const {
        return P(*this) *= a;
    }
    P operator/(const P &a) const {
        return P(*this) /= a;
    }
    P &operator+=(const P &r) {
        this->a.resize(max(this->a.size(), r.size()));
        for (int i = 0; i < (int)r.size(); i++) this->a[i] += r.a[i];
        return *this;
    }
 
    P &operator+=(const long long v) {
        if (this->a.size() == 0) this->a.resize(1, (v % MOD + MOD) % MOD);
        else
            this->a[0] += v;
        return *this;
    }
    P &operator-=(const P &r) {
        this->a.resize(max(this->a.size(), r.size()));
        for (int i = 0; i < (int)r.size(); i++) this->a[i] -= r.a[i];
        return *this;
    }
    P &operator*=(const P &b) {
        if (!any) {
            this->a = convolution(this->a, b.a);
            return *this;
        } else {
            if (this->a.empty() || b.a.empty()) {
                this->a.clear();
                return *this;
            }
            int n = this->a.size(), m = b.a.size();
            static constexpr int mod0 = 998244353, mod1 = 1300234241, mod2 = 1484783617;
            using Mint0 = mint< mod0 >;
            using Mint1 = mint< mod1 >;
            using Mint2 = mint< mod2 >;
            vector< Mint0 > l0(n), r0(m);
            vector< Mint1 > l1(n), r1(m);
            vector< Mint2 > l2(n), r2(m);
            for (int i = 0; i < n; i++) l0[i] = this->a[i].x, l1[i] = this->a[i].x, l2[i] = this->a[i].x;
            for (int j = 0; j < m; j++) r0[j] = b.a[j].x, r1[j] = b.a[j].x, r2[j] = b.a[j].x;
            l0 = convolution(l0, r0);
            l1 = convolution(l1, r1);
            l2 = convolution(l2, r2);
            this->a.resize(n + m - 1);
            static const Mint1 im0 = Mint1(1) / Mint1(mod0);
            static const Mint2 im1 = Mint2(1) / Mint2(mod1), im0m1 = im1 / mod0;
            static const mint< MOD > m0 = mod0, m0m1 = m0 * mod1;
            for (int i = 0; i < n + m - 1; i++) {
                int y0 = l0[i].x;
                int y1 = (im0 * (l1[i] - y0)).x;
                int y2 = (im0m1 * (l2[i] - y0) - im1 * y1).x;
                this->a[i] = m0m1 * y2 + y0 + m0 * y1;
            }
            return *this;
        }
    }
    P &operator*=(const long long v) {
        for (int i = 0; i < this->a.size(); i++) this->a[i] *= v;
        return *this;
    }
 
    P &operator/=(const P &a) {
        *this *= a.inverse();
        return *this;
    }
   //逆数
    P inverse(int deg = -1) const {
        assert(this->a.size() != 0 && this->a[0].x != 0);
        const int n = (int)this->a.size();
        if (deg == -1) deg = n;
        P ret(1);
        ret[0] = mint< MOD >(1) / a[0];
        for (int i = 1; i < deg; i <<= 1) {
            ret = pre((ret + ret - ret * ret * pre(*this, i << 1)), i << 1);
        }
        return pre(ret, deg);
    }
   //微分
    P differential() const {
        const int n = (int)this->a.size();
        P ret(max(0, n - 1));
        for (int i = 1; i < n; i++) ret[i - 1] = this->a[i] * i;
        return ret;
    }
    //不定積分
    P integral() const {
        const int n = (int)this->a.size();
        P ret(n + 1);
        for (int i = 0; i < n; i++) ret[i + 1] = this->a[i] / (i + 1);
        return ret;
    }
    P log(int deg = -1) const {
        assert(this->a.size() != 0 && this->a[0] == 1);
        const int n = (int)this->a.size();
        if (deg == -1) deg = n;
        return pre((this->differential() * this->inverse(deg)), deg - 1).integral();
    }
 
    P exp(int deg = -1) const {
        if (this->a.size() == 0) return P(0);
        assert(this->a[0] == 0);
        const int n = (int)this->a.size();
        if (deg == -1) deg = n;
        P ret(1);
        ret.a[0] = 1;
        for (int i = 1; i < deg; i <<= 1) {
            ret = pre((ret * (pre(*this, i << 1) + 1 - ret.log(i << 1))), i << 1);
        }
        return pre(ret, deg);
    }
 
    P pow(long long k, int deg = -1) const {
        const int n = (int)this->a.size();
        if (deg == -1) deg = n;
        for (int i = 0; i < n; i++) {
            if (this->a[i].x != 0) {
                long long rev = (mint< MOD >(1) / this->a[i]).x;
                P C = *this * rev;
                P D(n - i);
                for (int j = i; j < n; j++) D[j - i] = C[j];
                D = (D.log() * k).exp() * power(this->a[i], k).x;
                P E(deg);
                if (i * k > deg) return E;
                auto S = i * k;
                for (int j = 0; j + S < deg && j < D.size(); j++) E[j + S] = D[j];
                return E;
            }
        }
        return *this;
    }
    mint< MOD > &operator[](int x) {
        assert(0 <= x && x < (int)this->a.size());
        return a[x];
    }
    friend std::ostream &operator<<(std::ostream &os, const P &p) {
        os << "[ ";
        for (int i = 0; i < p.size(); ++i) {
            os << p.a[i] << " ";
        }
        os << "]";
        return os;
    }
};
//[x^k] p(x)/q(x)
template < int MOD, bool any = false >
mint< MOD > Bostan_Mori(FormalPowerSeries< MOD, any > p, FormalPowerSeries< MOD, any > q, ll k) {
    if(k<0)return 0;
    p.a.resize(max(p.size(), q.size()));
    q.a.resize(max(p.size(), q.size()));
    typedef FormalPowerSeries< MOD, any > FPS;
    while (k) {
        FPS _q = q;
        for (int i = 1; i < _q.size(); i += 2) _q[i] = -_q[i];
        FPS v = q, u = p;
        v *= _q;
        u *= _q;
        for (int i = k % 2; i < u.size(); i += 2) p[i / 2] = u[i];
        for (int i = 0; i < v.size(); i += 2) q[i / 2] = v[i];
        k /= 2;
    }
    return p[0] / q[0];
}

typedef FormalPowerSeries<mod,true> FPS;
 
int main(){
    ios::sync_with_stdio(false);cin.tie(0);cout.tie(0);
    int k;cin>>k;
    int n;cin>>n;
    int x[n];
    FPS f(100001);
    rep(i,0,n){
        cin>>x[i];
        f[x[i]]+=1;
    }
    cout<<Bostan_Mori({1},FPS{1}-f,k)<<endl;
    /*f=FPS{1}-f;
    f=FPS{1}/f;
    cout<<f[k]<<endl;*/
}


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