#pragma region Macros
#ifdef noimi
#pragma comment(linker, "/stack:256000000")
#include "my_template.hpp"
#else
//#pragma GCC target("avx2")
#pragma GCC optimize("Ofast")
#pragma GCC optimize("unroll-loops")

#include <immintrin.h>

#include <algorithm>
#include <array>
#include <bitset>
#include <cassert>
#include <cctype>
#include <cfenv>
#include <cfloat>
#include <chrono>
#include <cinttypes>
#include <climits>
#include <cmath>
#include <complex>
#include <cstdarg>
#include <cstddef>
#include <cstdint>
#include <cstdio>
#include <cstdlib>
#include <cstring>
#include <deque>
#include <fstream>
#include <functional>
#include <immintrin.h>
#include <initializer_list>
#include <iomanip>
#include <ios>
#include <iostream>
#include <istream>
#include <iterator>
#include <limits>
#include <list>
#include <map>
#include <memory>
#include <new>
#include <numeric>
#include <ostream>
#include <queue>
#include <random>
#include <set>
#include <sstream>
#include <stack>
#include <streambuf>
#include <string>
#include <tuple>
#include <type_traits>
#include <utility>
#include <variant>

#ifdef noimi
#define oj_local(a, b) b
#else
#define oj_local(a, b) a
#endif

#define LOCAL if(oj_local(0, 1))
#define OJ if(oj_local(1, 0))

using namespace std;
using ll = long long;
using ull = unsigned long long int;
using i128 = __int128_t;
using pii = pair<int, int>;
using pll = pair<ll, ll>;
using ld = long double;
template <typename T> using vc = vector<T>;
template <typename T> using vvc = vector<vc<T>>;
template <typename T> using vvvc = vector<vvc<T>>;
using vi = vc<int>;
using vl = vc<ll>;
using vpi = vc<pii>;
using vpl = vc<pll>;
template <class T> using pq = priority_queue<T>;
template <class T> using pqg = priority_queue<T, vector<T>, greater<T>>;
template <typename T> int si(const T &x) { return x.size(); }
template <class T, class S> inline bool chmax(T &a, const S &b) { return (a < b ? a = b, 1 : 0); }
template <class T, class S> inline bool chmin(T &a, const S &b) { return (a > b ? a = b, 1 : 0); }
vi iota(int n) {
    vi a(n);
    return iota(a.begin(), a.end(), 0), a;
}
template <typename T> vi iota(const vector<T> &a, bool greater = false) {
    vi res(a.size());
    iota(res.begin(), res.end(), 0);
    sort(res.begin(), res.end(), [&](int i, int j) {
        if(greater) return a[i] > a[j];
        return a[i] < a[j];
    });
    return res;
}

// macros
#define overload5(a, b, c, d, e, name, ...) name
#define overload4(a, b, c, d, name, ...) name
#define endl '\n'
#define REP0(n) for(ll jidlsjf = 0; jidlsjf < n; ++jidlsjf)
#define REP1(i, n) for(ll i = 0; i < (n); ++i)
#define REP2(i, a, b) for(ll i = (a); i < (b); ++i)
#define REP3(i, a, b, c) for(ll i = (a); i < (b); i += (c))
#define rep(...) overload4(__VA_ARGS__, REP3, REP2, REP1, REP0)(__VA_ARGS__)
#define per0(n) for(int jidlsjf = 0; jidlsjf < (n); ++jidlsjf)
#define per1(i, n) for(ll i = (n)-1; i >= 0; --i)
#define per2(i, a, b) for(ll i = (a)-1; i >= b; --i)
#define per3(i, a, b, c) for(ll i = (a)-1; i >= (b); i -= (c))
#define per(...) overload4(__VA_ARGS__, per3, per2, per1, per0)(__VA_ARGS__)
#define fore0(a) rep(a.size())
#define fore1(i, a) for(auto &&i : a)
#define fore2(a, b, v) for(auto &&[a, b] : v)
#define fore3(a, b, c, v) for(auto &&[a, b, c] : v)
#define fore4(a, b, c, d, v) for(auto &&[a, b, c, d] : v)
#define fore(...) overload5(__VA_ARGS__, fore4, fore3, fore2, fore1, fore0)(__VA_ARGS__)
#define setbits(j, n) for(ll iiiii = (n), j = lowbit(iiiii); iiiii; iiiii ^= 1 << j, j = lowbit(iiiii))
#define perm(v) for(bool permrepflag = true; (permrepflag ? exchange(permrepflag, false) : next_permutation(all(v)));)
#define fi first
#define se second
#define pb push_back
#define ppb pop_back
#define ppf pop_front
#define eb emplace_back
#define drop(s) cout << #s << endl, exit(0)
#define si(c) (int)(c).size()
#define lb(c, x) distance((c).begin(), lower_bound(all(c), (x)))
#define lbg(c, x) distance((c).begin(), lower_bound(all(c), (x), greater{}))
#define ub(c, x) distance((c).begin(), upper_bound(all(c), (x)))
#define ubg(c, x) distance((c).begin(), upper_bound(all(c), (x), greater{}))
#define rng(v, l, r) v.begin() + (l), v.begin() + (r)
#define all(c) begin(c), end(c)
#define rall(c) rbegin(c), rend(c)
#define SORT(v) sort(all(v))
#define REV(v) reverse(all(v))
#define UNIQUE(x) SORT(x), x.erase(unique(all(x)), x.end())
template <typename T = ll, typename S> T SUM(const S &v) { return accumulate(all(v), T(0)); }
#define MIN(v) *min_element(all(v))
#define MAX(v) *max_element(all(v))
#define overload2(_1, _2, name, ...) name
#define vec(type, name, ...) vector<type> name(__VA_ARGS__)
#define vv(type, name, h, ...) vector<vector<type>> name(h, vector<type>(__VA_ARGS__))
#define vvv(type, name, h, w, ...) vector<vector<vector<type>>> name(h, vector<vector<type>>(w, vector<type>(__VA_ARGS__)))
#define vvvv(type, name, a, b, c, ...)                                                                                                                         \
    vector<vector<vector<vector<type>>>> name(a, vector<vector<vector<type>>>(b, vector<vector<type>>(c, vector<type>(__VA_ARGS__))))
constexpr pii dx4[4] = {pii{1, 0}, pii{0, 1}, pii{-1, 0}, pii{0, -1}};
constexpr pii dx8[8] = {{1, 0}, {1, 1}, {0, 1}, {-1, 1}, {-1, 0}, {-1, -1}, {0, -1}, {1, -1}};

namespace yesno_impl {
const string YESNO[2] = {"NO", "YES"};
const string YesNo[2] = {"No", "Yes"};
const string yesno[2] = {"no", "yes"};
const string firstsecond[2] = {"second", "first"};
const string FirstSecond[2] = {"Second", "First"};
const string possiblestr[2] = {"impossible", "possible"};
const string Possiblestr[2] = {"Impossible", "Possible"};
void YES(bool t = 1) { cout << YESNO[t] << endl; }
void NO(bool t = 1) { YES(!t); }
void Yes(bool t = 1) { cout << YesNo[t] << endl; }
void No(bool t = 1) { Yes(!t); }
void yes(bool t = 1) { cout << yesno[t] << endl; }
void no(bool t = 1) { yes(!t); }
void first(bool t = 1) { cout << firstsecond[t] << endl; }
void First(bool t = 1) { cout << FirstSecond[t] << endl; }
void possible(bool t = 1) { cout << possiblestr[t] << endl; }
void Possible(bool t = 1) { cout << Possiblestr[t] << endl; }
}; // namespace yesno_impl
using namespace yesno_impl;

#define INT(...)                                                                                                                                               \
    int __VA_ARGS__;                                                                                                                                           \
    IN(__VA_ARGS__)
#define INTd(...)                                                                                                                                              \
    int __VA_ARGS__;                                                                                                                                           \
    IN2(__VA_ARGS__)
#define LL(...)                                                                                                                                                \
    ll __VA_ARGS__;                                                                                                                                            \
    IN(__VA_ARGS__)
#define LLd(...)                                                                                                                                               \
    ll __VA_ARGS__;                                                                                                                                            \
    IN2(__VA_ARGS__)
#define STR(...)                                                                                                                                               \
    string __VA_ARGS__;                                                                                                                                        \
    IN(__VA_ARGS__)
#define CHR(...)                                                                                                                                               \
    char __VA_ARGS__;                                                                                                                                          \
    IN(__VA_ARGS__)
#define DBL(...)                                                                                                                                               \
    double __VA_ARGS__;                                                                                                                                        \
    IN(__VA_ARGS__)
#define VEC(type, name, size)                                                                                                                                  \
    vector<type> name(size);                                                                                                                                   \
    IN(name)
#define VECd(type, name, size)                                                                                                                                 \
    vector<type> name(size);                                                                                                                                   \
    IN2(name)
#define VEC2(type, name1, name2, size)                                                                                                                         \
    vector<type> name1(size), name2(size);                                                                                                                     \
    for(int i = 0; i < size; i++) IN(name1[i], name2[i])
#define VEC2d(type, name1, name2, size)                                                                                                                        \
    vector<type> name1(size), name2(size);                                                                                                                     \
    for(int i = 0; i < size; i++) IN2(name1[i], name2[i])
#define VEC3(type, name1, name2, name3, size)                                                                                                                  \
    vector<type> name1(size), name2(size), name3(size);                                                                                                        \
    for(int i = 0; i < size; i++) IN(name1[i], name2[i], name3[i])
#define VEC3d(type, name1, name2, name3, size)                                                                                                                 \
    vector<type> name1(size), name2(size), name3(size);                                                                                                        \
    for(int i = 0; i < size; i++) IN2(name1[i], name2[i], name3[i])
#define VEC4(type, name1, name2, name3, name4, size)                                                                                                           \
    vector<type> name1(size), name2(size), name3(size), name4(size);                                                                                           \
    for(int i = 0; i < size; i++) IN(name1[i], name2[i], name3[i], name4[i]);
#define VEC4d(type, name1, name2, name3, name4, size)                                                                                                          \
    vector<type> name1(size), name2(size), name3(size), name4(size);                                                                                           \
    for(int i = 0; i < size; i++) IN2(name1[i], name2[i], name3[i], name4[i]);
#define VV(type, name, h, w)                                                                                                                                   \
    vector<vector<type>> name(h, vector<type>(w));                                                                                                             \
    IN(name)
#define VVd(type, name, h, w)                                                                                                                                  \
    vector<vector<type>> name(h, vector<type>(w));                                                                                                             \
    IN2(name)
int scan() { return getchar(); }
void scan(int &a) { cin >> a; }
void scan(long long &a) { cin >> a; }
void scan(char &a) { cin >> a; }
void scan(double &a) { cin >> a; }
void scan(string &a) { cin >> a; }
template <class T, class S> void scan(pair<T, S> &p) { scan(p.first), scan(p.second); }
template <class T> void scan(vector<T> &);
template <class T> void scan(vector<T> &a) {
    for(auto &i : a) scan(i);
}
template <class T> void scan(T &a) { cin >> a; }
void IN() {}
void IN2() {}
template <class Head, class... Tail> void IN(Head &head, Tail &...tail) {
    scan(head);
    IN(tail...);
}
template <class Head, class... Tail> void IN2(Head &head, Tail &...tail) {
    scan(head);
    --head;
    IN2(tail...);
}

template <int p = -1> void pat() {}
template <int p = -1, class Head, class... Tail> void pat(Head &h, Tail &...tail) {
    h += p;
    pat<p>(tail...);
}

template <typename T, typename S> T ceil(T x, S y) {
    assert(y);
    return (y < 0 ? ceil(-x, -y) : (x > 0 ? (x + y - 1) / y : x / y));
}

template <typename T, typename S> T floor(T x, S y) {
    assert(y);
    return (y < 0 ? floor(-x, -y) : (x > 0 ? x / y : x / y - (x % y == 0 ? 0 : 1)));
}
template <typename T, typename S, typename U> U bigmul(const T &x, const S &y, const U &lim) { // clamp(x * y, -lim, lim)
    if(x < 0 and y < 0) return bigmul(-x, -y, lim);
    if(x < 0) return -bigmul(-x, y, lim);
    if(y < 0) return -bigmul(x, -y, lim);
    return y == 0 or x <= lim / y ? x * y : lim;
}
template <class T> T POW(T x, int n) {
    T res = 1;
    for(; n; n >>= 1, x *= x)
        if(n & 1) res *= x;
    return res;
}
template <class T, class S> T POW(T x, S n, const ll &mod) {
    T res = 1;
    x %= mod;
    for(; n; n >>= 1, x = x * x % mod)
        if(n & 1) res = res * x % mod;
    return res;
}
vector<pll> factor(ll x) {
    vector<pll> ans;
    for(ll i = 2; i * i <= x; i++)
        if(x % i == 0) {
            ans.push_back({i, 1});
            while((x /= i) % i == 0) ans.back().second++;
        }
    if(x != 1) ans.push_back({x, 1});
    return ans;
}
template <class T> vector<T> divisor(T x) {
    vector<T> ans;
    for(T i = 1; i * i <= x; i++)
        if(x % i == 0) {
            ans.pb(i);
            if(i * i != x) ans.pb(x / i);
        }
    return ans;
}
template <typename T> void zip(vector<T> &x) {
    vector<T> y = x;
    UNIQUE(y);
    for(int i = 0; i < x.size(); ++i) { x[i] = lb(y, x[i]); }
}
template <class S> void fold_in(vector<S> &v) {}
template <typename Head, typename... Tail, class S> void fold_in(vector<S> &v, Head &&a, Tail &&...tail) {
    for(auto e : a) v.emplace_back(e);
    fold_in(v, tail...);
}
template <class S> void renumber(vector<S> &v) {}
template <typename Head, typename... Tail, class S> void renumber(vector<S> &v, Head &&a, Tail &&...tail) {
    for(auto &&e : a) e = lb(v, e);
    renumber(v, tail...);
}
template <class S, class... Args> vector<S> zip(vector<S> &head, Args &&...args) {
    vector<S> v;
    fold_in(v, head, args...);
    sort(all(v)), v.erase(unique(all(v)), v.end());
    renumber(v, head, args...);
    return v;
}

template <typename S> void rearrange(const vector<S> &id) {}
template <typename S, typename T> void rearrange_exec(const vector<S> &id, vector<T> &v) {
    vector<T> w(v.size());
    rep(i, si(id)) w[i] = v[id[i]];
    v.swap(w);
}
// 並び替える順番, 並び替える vector 達
template <typename S, typename Head, typename... Tail> void rearrange(const vector<S> &id, Head &a, Tail &...tail) {
    rearrange_exec(id, a);
    rearrange(id, tail...);
}

template <typename T> vector<T> RUI(const vector<T> &v) {
    vector<T> res(v.size() + 1);
    for(int i = 0; i < v.size(); i++) res[i + 1] = res[i] + v[i];
    return res;
}
template <typename T> void zeta_supersetsum(vector<T> &f) {
    int n = f.size();
    for(int i = 1; i < n; i <<= 1) rep(b, n) if(!(i & b)) f[b] += f[b | i];
}

template <typename T> void zeta_subsetsum(vector<T> &f) {
    int n = f.size();
    for(int i = 1; i < n; i <<= 1) rep(b, n) if(!(i & b)) f[b | i] += f[b];
}
template <typename T> void mobius_subset(vector<T> &f) {
    int n = f.size();
    for(int i = 1; i < n; i <<= 1) rep(b, n) if(!(i & b)) f[b] -= f[b | i];
}
template <typename T> void mobius_superset(vector<T> &f) {
    int n = f.size();
    for(int i = 1; i < n; i <<= 1) rep(b, n) if(!(i & b)) f[b | i] -= f[b];
}
// 反時計周りに 90 度回転
template <typename T> void rot(vector<vector<T>> &v) {
    if(empty(v)) return;
    int n = v.size(), m = v[0].size();
    vector<vector<T>> res(m, vector<T>(n));
    rep(i, n) rep(j, m) res[m - 1 - j][i] = v[i][j];
    v.swap(res);
}

vector<int> counter(const vector<int> &v, int max_num = -1) {
    if(max_num == -1) max_num = MAX(v);
    vector<int> res(max_num + 1);
    fore(e, v) res[e]++;
    return res;
}

// x in [l, r)
template <class T, class S> bool inc(const T &x, const S &l, const S &r) { return l <= x and x < r; }
template <class T, class S> bool inc(const T &x, const pair<S, S> &p) { return p.first <= x and x < p.second; }

// 便利関数
constexpr ll ten(int n) { return n == 0 ? 1 : ten(n - 1) * 10; }
constexpr ll tri(ll n) { return n * (n + 1) / 2; }
// l + ... + r
constexpr ll tri(ll l, ll r) { return (l + r) * (r - l + 1) / 2; }
ll max(int x, ll y) { return max((ll)x, y); }
ll max(ll x, int y) { return max(x, (ll)y); }
int min(int x, ll y) { return min((ll)x, y); }
int min(ll x, int y) { return min(x, (ll)y); }
// bit 演算系
#define bit(i) (1LL << i)       // (1 << i)
#define test(b, i) (b >> i & 1) // b の i bit 目が立っているか
ll pow2(int i) { return 1LL << i; }
int topbit(signed t) { return t == 0 ? -1 : 31 - __builtin_clz(t); }
int topbit(ll t) { return t == 0 ? -1 : 63 - __builtin_clzll(t); }
// int lowbit(signed a) { return a == 0 ? 32 : __builtin_ctz(a); }
int lowbit(ull a) { return a == 0 ? 64 : __builtin_ctzll(a); }
// int allbit(int n) { return (1 << n) - 1; }
constexpr ll mask(int n) { return (1LL << n) - 1; }
// int popcount(signed t) { return __builtin_popcount(t); }
// int popcount(ll t) { return __builtin_popcountll(t); }
int popcount(uint64_t t) { return __builtin_popcountll(t); }
static inline uint64_t popcount64(uint64_t x) {
    uint64_t m1 = 0x5555555555555555ll;
    uint64_t m2 = 0x3333333333333333ll;
    uint64_t m4 = 0x0F0F0F0F0F0F0F0Fll;
    uint64_t h01 = 0x0101010101010101ll;

    x -= (x >> 1) & m1;
    x = (x & m2) + ((x >> 2) & m2);
    x = (x + (x >> 4)) & m4;

    return (x * h01) >> 56;
}
bool ispow2(int i) { return i && (i & -i) == i; }

ll rnd(ll l, ll r) { //[l, r)
#ifdef noimi
    static mt19937_64 gen;
#else
    static mt19937_64 gen(chrono::steady_clock::now().time_since_epoch().count());
#endif
    return uniform_int_distribution<ll>(l, r - 1)(gen);
}
ll rnd(ll n) { return rnd(0, n); }

template <class t> void random_shuffle(vc<t> &a) { rep(i, si(a)) swap(a[i], a[rnd(0, i + 1)]); }

int in() {
    int x;
    cin >> x;
    return x;
}
ll lin() {
    unsigned long long x;
    cin >> x;
    return x;
}

template <class T, class S> pair<T, S> operator-(const pair<T, S> &x) { return pair<T, S>(-x.first, -x.second); }
template <class T, class S> pair<T, S> operator-(const pair<T, S> &x, const pair<T, S> &y) { return pair<T, S>(x.fi - y.fi, x.se - y.se); }
template <class T, class S> pair<T, S> operator+(const pair<T, S> &x, const pair<T, S> &y) { return pair<T, S>(x.fi + y.fi, x.se + y.se); }
template <class T> pair<T, T> operator&(const pair<T, T> &l, const pair<T, T> &r) { return pair<T, T>(max(l.fi, r.fi), min(l.se, r.se)); }
template <class T, class S> pair<T, S> operator+=(pair<T, S> &l, const pair<T, S> &r) { return l = l + r; }
template <class T, class S> pair<T, S> operator-=(pair<T, S> &l, const pair<T, S> &r) { return l = l - r; }
template <class T> bool intersect(const pair<T, T> &l, const pair<T, T> &r) { return (l.se < r.se ? r.fi < l.se : l.fi < r.se); }

template <class T> vector<T> &operator++(vector<T> &v) {
    fore(e, v) e++;
    return v;
}
template <class T> vector<T> operator++(vector<T> &v, int) {
    auto res = v;
    fore(e, v) e++;
    return res;
}
template <class T> vector<T> &operator--(vector<T> &v) {
    fore(e, v) e--;
    return v;
}
template <class T> vector<T> operator--(vector<T> &v, int) {
    auto res = v;
    fore(e, v) e--;
    return res;
}
template <class T> void connect(vector<T> &l, const vector<T> &r) { fore(e, r) l.eb(e); }
template <class T> vector<T> operator+(const vector<T> &l, const vector<T> &r) {
    vector<T> res(max(si(l), si(r)));
    rep(i, si(l)) res[i] += l[i];
    rep(i, si(r)) res[i] += r[i];
    return res;
}
template <class T> vector<T> operator-(const vector<T> &l, const vector<T> &r) {
    vector<T> res(max(si(l), si(r)));
    rep(i, si(l)) res[i] += l[i];
    rep(i, si(r)) res[i] -= r[i];
    return res;
}
template <class T> vector<T> &operator+=(const vector<T> &l, const vector<T> &r) {
    if(si(l) < si(r)) l.resize(si(r));
    rep(i, si(r)) l[i] += r[i];
    return l;
}
template <class T> vector<T> &operator-=(const vector<T> &l, const vector<T> &r) {
    if(si(l) < si(r)) l.resize(si(r));
    rep(i, si(r)) l[i] -= r[i];
    return l;
}
template <class T> vector<T> &operator+=(vector<T> &v, const T &x) {
    fore(e, v) e += x;
    return v;
}
template <class T> vector<T> &operator-=(vector<T> &v, const T &x) {
    fore(e, v) e -= x;
    return v;
}

template <typename T> struct edge {
    int from, to;
    T cost;
    int id;
    edge(int to, T cost) : from(-1), to(to), cost(cost) {}
    edge(int from, int to, T cost) : from(from), to(to), cost(cost) {}
    edge(int from, int to, T cost, int id) : from(from), to(to), cost(cost), id(id) {}
    constexpr bool operator<(const edge<T> &rhs) const noexcept { return cost < rhs.cost; }
    edge &operator=(const int &x) {
        to = x;
        return *this;
    }
    operator int() const { return to; }
    friend ostream operator<<(ostream &os, const edge &e) { return os << e.to; }
};
template <typename T> using Edges = vector<edge<T>>;

template <typename T = int> Edges<T> read_edges(int m, bool weighted = false) {
    Edges<T> res;
    res.reserve(m);
    for(int i = 0; i < m; i++) {
        int u, v, c = 0;
        scan(u), scan(v), u--, v--;
        if(weighted) scan(c);
        res.eb(u, v, c, i);
    }
    return res;
}

using Tree = vector<vector<int>>;
using Graph = vector<vector<int>>;
template <class T> using Wgraph = vector<vector<edge<T>>>;
Graph getG(int n, int m = -1, bool directed = false, int margin = 1) {
    Tree res(n);
    if(m == -1) m = n - 1;
    while(m--) {
        int a, b;
        cin >> a >> b;
        a -= margin, b -= margin;
        res[a].emplace_back(b);
        if(!directed) res[b].emplace_back(a);
    }
    return res;
}
Graph getTreeFromPar(int n, int margin = 1) {
    Graph res(n);
    for(int i = 1; i < n; i++) {
        int a;
        cin >> a;
        res[a - margin].emplace_back(i);
    }
    return res;
}
template <class T> Wgraph<T> getWg(int n, int m = -1, bool directed = false, int margin = 1) {
    Wgraph<T> res(n);
    if(m == -1) m = n - 1;
    while(m--) {
        int a, b;
        T c;
        scan(a), scan(b), scan(c);
        a -= margin, b -= margin;
        res[a].emplace_back(b, c);
        if(!directed) res[b].emplace_back(a, c);
    }
    return res;
}
void add(Graph &G, int x, int y) { G[x].eb(y), G[y].eb(x); }
template <class S, class T> void add(Wgraph<S> &G, int x, int y, T c) { G[x].eb(y, c), G[y].eb(x, c); }

#define TEST                                                                                                                                                   \
    INT(testcases);                                                                                                                                            \
    while(testcases--)

i128 abs(const i128 &x) { return x > 0 ? x : -x; }
istream &operator>>(istream &is, i128 &v) {
    string s;
    is >> s;
    v = 0;
    for(int i = 0; i < (int)s.size(); i++) {
        if(isdigit(s[i])) { v = v * 10 + s[i] - '0'; }
    }
    if(s[0] == '-') { v *= -1; }
    return is;
}

ostream &operator<<(ostream &os, const i128 &v) {
    if(v == 0) { return (os << "0"); }
    i128 num = v;
    if(v < 0) {
        os << '-';
        num = -num;
    }
    string s;
    for(; num > 0; num /= 10) { s.push_back((char)(num % 10) + '0'); }
    reverse(s.begin(), s.end());
    return (os << s);
}
namespace aux {
template <typename T, unsigned N, unsigned L> struct tp {
    static void output(std::ostream &os, const T &v) {
        os << std::get<N>(v) << (&os == &cerr ? ", " : " ");
        tp<T, N + 1, L>::output(os, v);
    }
};
template <typename T, unsigned N> struct tp<T, N, N> {
    static void output(std::ostream &os, const T &v) { os << std::get<N>(v); }
};
} // namespace aux
template <typename... Ts> std::ostream &operator<<(std::ostream &os, const std::tuple<Ts...> &t) {
    if(&os == &cerr) { os << '('; }
    aux::tp<std::tuple<Ts...>, 0, sizeof...(Ts) - 1>::output(os, t);
    if(&os == &cerr) { os << ')'; }
    return os;
}
template <typename T, typename S, typename U> std::ostream &operator<<(std::ostream &os, const priority_queue<T, S, U> &_pq) {
    auto pq = _pq;
    vector<T> res;
    while(!empty(pq)) res.emplace_back(pq.top()), pq.pop();
    return os << res;
}
template <class T, class S> ostream &operator<<(ostream &os, const pair<T, S> &p) {
    if(&os == &cerr) { return os << "(" << p.first << ", " << p.second << ")"; }
    return os << p.first << " " << p.second;
}
template <class Ch, class Tr, class Container> std::basic_ostream<Ch, Tr> &operator<<(std::basic_ostream<Ch, Tr> &os, const Container &x) {
    bool f = true;
    if(&os == &cerr) os << "[";
    for(auto &y : x) {
        if(&os == &cerr)
            os << (f ? "" : ", ") << y;
        else
            os << (f ? "" : " ") << y;
        f = false;
    }
    if(&os == &cerr) os << "]";
    return os;
}

#define dump(...) static_cast<void>(0)
#define dbg(...) static_cast<void>(0)

void OUT() { cout << endl; }
template <class Head, class... Tail> void OUT(const Head &head, const Tail &...tail) {
    cout << head;
    if(sizeof...(tail)) cout << ' ';
    OUT(tail...);
}

template <typename T> static constexpr T inf = numeric_limits<T>::max() / 2;
template <class T, class S> constexpr pair<T, S> inf<pair<T, S>> = {inf<T>, inf<S>};

template <class T> void OUT2(const T &t, T INF = inf<T>, T res = -1) { OUT(t != INF ? t : res); }
template <class T> void OUT2(vector<T> &v, T INF = inf<T>, T res = -1) {
    fore(e, v) if(e == INF) e = res;
    OUT(v);
    fore(e, v) if(e == res) e = INF;
}

template <class F> struct REC {
    F f;
    REC(F &&f_) : f(forward<F>(f_)) {}
    template <class... Args> auto operator()(Args &&...args) const { return f(*this, forward<Args>(args)...); }
};

template <class S> vector<pair<S, int>> runLength(const vector<S> &v) {
    vector<pair<S, int>> res;
    for(auto &e : v) {
        if(res.empty() or res.back().fi != e)
            res.eb(e, 1);
        else
            res.back().se++;
    }
    return res;
}
vector<pair<char, int>> runLength(const string &v) {
    vector<pair<char, int>> res;
    for(auto &e : v) {
        if(res.empty() or res.back().fi != e)
            res.eb(e, 1);
        else
            res.back().se++;
    }
    return res;
}

struct string_converter {
    char start = 0;
    char type(const char &c) const { return (islower(c) ? 'a' : isupper(c) ? 'A' : isdigit(c) ? '0' : 0); }
    int convert(const char &c) {
        if(!start) start = type(c);
        return c - start;
    }
    int convert(const char &c, const string &chars) { return chars.find(c); }
    template <typename T> auto convert(const T &v) {
        vector<decltype(convert(v[0]))> ret;
        ret.reserve(size(v));
        for(auto &&e : v) ret.emplace_back(convert(e));
        return ret;
    }
    template <typename T> auto convert(const T &v, const string &chars) {
        vector<decltype(convert(v[0], chars))> ret;
        ret.reserve(size(v));
        for(auto &&e : v) ret.emplace_back(convert(e, chars));
        return ret;
    }
    int operator()(const char &v, char s = 0) {
        start = s;
        return convert(v);
    }
    int operator()(const char &v, const string &chars) { return convert(v, chars); }
    template <typename T> auto operator()(const T &v, char s = 0) {
        start = s;
        return convert(v);
    }
    template <typename T> auto operator()(const T &v, const string &chars) { return convert(v, chars); }
} toint;

template <class T, class F> T bin_search(T ok, T ng, const F &f) {
    while(abs(ok - ng) > 1) {
        T mid = ok + ng >> 1;
        (f(mid) ? ok : ng) = mid;
    }
    return ok;
}
template <class T, class F> T bin_search_double(T ok, T ng, const F &f, int iter = 80) {
    while(iter--) {
        T mid = (ok + ng) / 2;
        (f(mid) ? ok : ng) = mid;
    }
    return ok;
}

struct Setup_io {
    Setup_io() {
        ios_base::sync_with_stdio(0), cin.tie(0), cout.tie(0);
        cout << fixed << setprecision(11);
    }
} setup_io;

#endif

#pragma endregion

template <uint32_t mod> struct LazyMontgomeryModInt {
    using mint = LazyMontgomeryModInt;
    using i32 = int32_t;
    using u32 = uint32_t;
    using u64 = uint64_t;

    static constexpr u32 get_r() {
        u32 ret = mod;
        for(i32 i = 0; i < 4; ++i) ret *= 2 - mod * ret;
        return ret;
    }

    static constexpr u32 r = get_r();
    static constexpr u32 n2 = -u64(mod) % mod;
    static_assert(mod < (1 << 30), "invalid, mod >= 2 ^ 30");
    static_assert((mod & 1) == 1, "invalid, mod % 2 == 0");
    static_assert(r * mod == 1, "this code has bugs.");

    u32 a;

    constexpr LazyMontgomeryModInt() : a(0) {}
    constexpr LazyMontgomeryModInt(const int64_t &b) : a(reduce(u64(b % mod + mod) * n2)){};

    static constexpr u32 reduce(const u64 &b) { return (b + u64(u32(b) * u32(-r)) * mod) >> 32; }

    constexpr mint &operator+=(const mint &b) {
        if(i32(a += b.a - 2 * mod) < 0) a += 2 * mod;
        return *this;
    }

    constexpr mint &operator-=(const mint &b) {
        if(i32(a -= b.a) < 0) a += 2 * mod;
        return *this;
    }

    constexpr mint &operator*=(const mint &b) {
        a = reduce(u64(a) * b.a);
        return *this;
    }

    constexpr mint &operator/=(const mint &b) {
        *this *= b.inverse();
        return *this;
    }

    constexpr mint operator+(const mint &b) const { return mint(*this) += b; }
    constexpr mint operator-(const mint &b) const { return mint(*this) -= b; }
    constexpr mint operator*(const mint &b) const { return mint(*this) *= b; }
    constexpr mint operator/(const mint &b) const { return mint(*this) /= b; }
    constexpr bool operator==(const mint &b) const { return (a >= mod ? a - mod : a) == (b.a >= mod ? b.a - mod : b.a); }
    constexpr bool operator!=(const mint &b) const { return (a >= mod ? a - mod : a) != (b.a >= mod ? b.a - mod : b.a); }
    constexpr mint operator-() const { return mint() - mint(*this); }
    constexpr mint operator+() const { return mint(*this); }

    constexpr mint pow(u64 n) const {
        mint ret(1), mul(*this);
        while(n > 0) {
            if(n & 1) ret *= mul;
            mul *= mul;
            n >>= 1;
        }
        return ret;
    }

    constexpr mint inverse() const {
        int x = get(), y = mod, u = 1, v = 0, t = 0, tmp = 0;
        while(y > 0) {
            t = x / y;
            x -= t * y, u -= t * v;
            tmp = x, x = y, y = tmp;
            tmp = u, u = v, v = tmp;
        }
        return mint{u};
    }

    friend ostream &operator<<(ostream &os, const mint &b) { return os << b.get(); }

    friend istream &operator>>(istream &is, mint &b) {
        int64_t t;
        is >> t;
        b = LazyMontgomeryModInt<mod>(t);
        return (is);
    }

    constexpr u32 get() const {
        u32 ret = reduce(a);
        return ret >= mod ? ret - mod : ret;
    }

    static constexpr u32 get_mod() { return mod; }
};

template <typename mint> struct NTT {
    static constexpr uint32_t get_pr() {
        uint32_t _mod = mint::get_mod();
        using u64 = uint64_t;
        u64 ds[32] = {};
        int idx = 0;
        u64 m = _mod - 1;
        for(u64 i = 2; i * i <= m; ++i) {
            if(m % i == 0) {
                ds[idx++] = i;
                while(m % i == 0) m /= i;
            }
        }
        if(m != 1) ds[idx++] = m;

        uint32_t _pr = 2;
        while(1) {
            int flg = 1;
            for(int i = 0; i < idx; ++i) {
                u64 a = _pr, b = (_mod - 1) / ds[i], r = 1;
                while(b) {
                    if(b & 1) r = r * a % _mod;
                    a = a * a % _mod;
                    b >>= 1;
                }
                if(r == 1) {
                    flg = 0;
                    break;
                }
            }
            if(flg == 1) break;
            ++_pr;
        }
        return _pr;
    };

    static constexpr uint32_t mod = mint::get_mod();
    static constexpr uint32_t pr = get_pr();
    static constexpr int level = __builtin_ctzll(mod - 1);
    mint dw[level], dy[level];

    void setwy(int k) {
        mint w[level], y[level];
        w[k - 1] = mint(pr).pow((mod - 1) / (1 << k));
        y[k - 1] = w[k - 1].inverse();
        for(int i = k - 2; i > 0; --i) w[i] = w[i + 1] * w[i + 1], y[i] = y[i + 1] * y[i + 1];
        dw[1] = w[1], dy[1] = y[1], dw[2] = w[2], dy[2] = y[2];
        for(int i = 3; i < k; ++i) {
            dw[i] = dw[i - 1] * y[i - 2] * w[i];
            dy[i] = dy[i - 1] * w[i - 2] * y[i];
        }
    }

    NTT() { setwy(level); }

    void fft4(vector<mint> &a, int k) {
        if((int)a.size() <= 1) return;
        if(k == 1) {
            mint a1 = a[1];
            a[1] = a[0] - a[1];
            a[0] = a[0] + a1;
            return;
        }
        if(k & 1) {
            int v = 1 << (k - 1);
            for(int j = 0; j < v; ++j) {
                mint ajv = a[j + v];
                a[j + v] = a[j] - ajv;
                a[j] += ajv;
            }
        }
        int u = 1 << (2 + (k & 1));
        int v = 1 << (k - 2 - (k & 1));
        mint one = mint(1);
        mint imag = dw[1];
        while(v) {
            // jh = 0
            {
                int j0 = 0;
                int j1 = v;
                int j2 = j1 + v;
                int j3 = j2 + v;
                for(; j0 < v; ++j0, ++j1, ++j2, ++j3) {
                    mint t0 = a[j0], t1 = a[j1], t2 = a[j2], t3 = a[j3];
                    mint t0p2 = t0 + t2, t1p3 = t1 + t3;
                    mint t0m2 = t0 - t2, t1m3 = (t1 - t3) * imag;
                    a[j0] = t0p2 + t1p3, a[j1] = t0p2 - t1p3;
                    a[j2] = t0m2 + t1m3, a[j3] = t0m2 - t1m3;
                }
            }
            // jh >= 1
            mint ww = one, xx = one * dw[2], wx = one;
            for(int jh = 4; jh < u;) {
                ww = xx * xx, wx = ww * xx;
                int j0 = jh * v;
                int je = j0 + v;
                int j2 = je + v;
                for(; j0 < je; ++j0, ++j2) {
                    mint t0 = a[j0], t1 = a[j0 + v] * xx, t2 = a[j2] * ww, t3 = a[j2 + v] * wx;
                    mint t0p2 = t0 + t2, t1p3 = t1 + t3;
                    mint t0m2 = t0 - t2, t1m3 = (t1 - t3) * imag;
                    a[j0] = t0p2 + t1p3, a[j0 + v] = t0p2 - t1p3;
                    a[j2] = t0m2 + t1m3, a[j2 + v] = t0m2 - t1m3;
                }
                xx *= dw[__builtin_ctzll((jh += 4))];
            }
            u <<= 2;
            v >>= 2;
        }
    }

    void ifft4(vector<mint> &a, int k) {
        if((int)a.size() <= 1) return;
        if(k == 1) {
            mint a1 = a[1];
            a[1] = a[0] - a[1];
            a[0] = a[0] + a1;
            return;
        }
        int u = 1 << (k - 2);
        int v = 1;
        mint one = mint(1);
        mint imag = dy[1];
        while(u) {
            // jh = 0
            {
                int j0 = 0;
                int j1 = v;
                int j2 = v + v;
                int j3 = j2 + v;
                for(; j0 < v; ++j0, ++j1, ++j2, ++j3) {
                    mint t0 = a[j0], t1 = a[j1], t2 = a[j2], t3 = a[j3];
                    mint t0p1 = t0 + t1, t2p3 = t2 + t3;
                    mint t0m1 = t0 - t1, t2m3 = (t2 - t3) * imag;
                    a[j0] = t0p1 + t2p3, a[j2] = t0p1 - t2p3;
                    a[j1] = t0m1 + t2m3, a[j3] = t0m1 - t2m3;
                }
            }
            // jh >= 1
            mint ww = one, xx = one * dy[2], yy = one;
            u <<= 2;
            for(int jh = 4; jh < u;) {
                ww = xx * xx, yy = xx * imag;
                int j0 = jh * v;
                int je = j0 + v;
                int j2 = je + v;
                for(; j0 < je; ++j0, ++j2) {
                    mint t0 = a[j0], t1 = a[j0 + v], t2 = a[j2], t3 = a[j2 + v];
                    mint t0p1 = t0 + t1, t2p3 = t2 + t3;
                    mint t0m1 = (t0 - t1) * xx, t2m3 = (t2 - t3) * yy;
                    a[j0] = t0p1 + t2p3, a[j2] = (t0p1 - t2p3) * ww;
                    a[j0 + v] = t0m1 + t2m3, a[j2 + v] = (t0m1 - t2m3) * ww;
                }
                xx *= dy[__builtin_ctzll(jh += 4)];
            }
            u >>= 4;
            v <<= 2;
        }
        if(k & 1) {
            u = 1 << (k - 1);
            for(int j = 0; j < u; ++j) {
                mint ajv = a[j] - a[j + u];
                a[j] += a[j + u];
                a[j + u] = ajv;
            }
        }
    }

    void ntt(vector<mint> &a) {
        if((int)a.size() <= 1) return;
        fft4(a, __builtin_ctz(a.size()));
    }

    void intt(vector<mint> &a) {
        if((int)a.size() <= 1) return;
        ifft4(a, __builtin_ctz(a.size()));
        mint iv = mint(a.size()).inverse();
        for(auto &x : a) x *= iv;
    }

    vector<mint> multiply(const vector<mint> &a, const vector<mint> &b) {
        int l = a.size() + b.size() - 1;
        if(min<int>(a.size(), b.size()) <= 40) {
            vector<mint> s(l);
            for(int i = 0; i < (int)a.size(); ++i)
                for(int j = 0; j < (int)b.size(); ++j) s[i + j] += a[i] * b[j];
            return s;
        }
        int k = 2, M = 4;
        while(M < l) M <<= 1, ++k;
        setwy(k);
        vector<mint> s(M);
        for(int i = 0; i < (int)a.size(); ++i) s[i] = a[i];
        fft4(s, k);
        if(a.size() == b.size() && a == b) {
            for(int i = 0; i < M; ++i) s[i] *= s[i];
        } else {
            vector<mint> t(M);
            for(int i = 0; i < (int)b.size(); ++i) t[i] = b[i];
            fft4(t, k);
            for(int i = 0; i < M; ++i) s[i] *= t[i];
        }
        ifft4(s, k);
        s.resize(l);
        mint invm = mint(M).inverse();
        for(int i = 0; i < l; ++i) s[i] *= invm;
        return s;
    }

    void ntt_doubling(vector<mint> &a) {
        int M = (int)a.size();
        auto b = a;
        intt(b);
        mint r = 1, zeta = mint(pr).pow((mint::get_mod() - 1) / (M << 1));
        for(int i = 0; i < M; i++) b[i] *= r, r *= zeta;
        ntt(b);
        copy(begin(b), end(b), back_inserter(a));
    }
};

namespace ArbitraryNTT {
using i64 = int64_t;
using u128 = __uint128_t;
constexpr int32_t m0 = 167772161;
constexpr int32_t m1 = 469762049;
constexpr int32_t m2 = 754974721;
using mint0 = LazyMontgomeryModInt<m0>;
using mint1 = LazyMontgomeryModInt<m1>;
using mint2 = LazyMontgomeryModInt<m2>;
constexpr int r01 = mint1(m0).inverse().get();
constexpr int r02 = mint2(m0).inverse().get();
constexpr int r12 = mint2(m1).inverse().get();
constexpr int r02r12 = i64(r02) * r12 % m2;
constexpr i64 w1 = m0;
constexpr i64 w2 = i64(m0) * m1;

template <typename T, typename submint> vector<submint> mul(const vector<T> &a, const vector<T> &b) {
    static NTT<submint> ntt;
    vector<submint> s(a.size()), t(b.size());
    for(int i = 0; i < (int)a.size(); ++i) s[i] = i64(a[i] % submint::get_mod());
    for(int i = 0; i < (int)b.size(); ++i) t[i] = i64(b[i] % submint::get_mod());
    return ntt.multiply(s, t);
}

template <typename T> vector<int> multiply(const vector<T> &s, const vector<T> &t, int mod) {
    auto d0 = mul<T, mint0>(s, t);
    auto d1 = mul<T, mint1>(s, t);
    auto d2 = mul<T, mint2>(s, t);
    int n = d0.size();
    vector<int> ret(n);
    const int W1 = w1 % mod;
    const int W2 = w2 % mod;
    for(int i = 0; i < n; i++) {
        int n1 = d1[i].get(), n2 = d2[i].get(), a = d0[i].get();
        int b = i64(n1 + m1 - a) * r01 % m1;
        int c = (i64(n2 + m2 - a) * r02r12 + i64(m2 - b) * r12) % m2;
        ret[i] = (i64(a) + i64(b) * W1 + i64(c) * W2) % mod;
    }
    return ret;
}

template <typename mint> vector<mint> multiply(const vector<mint> &a, const vector<mint> &b) {
    if(a.size() == 0 && b.size() == 0) return {};
    if(min<int>(a.size(), b.size()) < 128) {
        vector<mint> ret(a.size() + b.size() - 1);
        for(int i = 0; i < (int)a.size(); ++i)
            for(int j = 0; j < (int)b.size(); ++j) ret[i + j] += a[i] * b[j];
        return ret;
    }
    vector<int> s(a.size()), t(b.size());
    for(int i = 0; i < (int)a.size(); ++i) s[i] = a[i].get();
    for(int i = 0; i < (int)b.size(); ++i) t[i] = b[i].get();
    vector<int> u = multiply<int>(s, t, mint::get_mod());
    vector<mint> ret(u.size());
    for(int i = 0; i < (int)u.size(); ++i) ret[i] = mint(u[i]);
    return ret;
}

template <typename T> vector<u128> multiply_u128(const vector<T> &s, const vector<T> &t) {
    if(s.size() == 0 && t.size() == 0) return {};
    if(min<int>(s.size(), t.size()) < 128) {
        vector<u128> ret(s.size() + t.size() - 1);
        for(int i = 0; i < (int)s.size(); ++i)
            for(int j = 0; j < (int)t.size(); ++j) ret[i + j] += i64(s[i]) * t[j];
        return ret;
    }
    auto d0 = mul<T, mint0>(s, t);
    auto d1 = mul<T, mint1>(s, t);
    auto d2 = mul<T, mint2>(s, t);
    int n = d0.size();
    vector<u128> ret(n);
    for(int i = 0; i < n; i++) {
        i64 n1 = d1[i].get(), n2 = d2[i].get();
        i64 a = d0[i].get();
        i64 b = (n1 + m1 - a) * r01 % m1;
        i64 c = ((n2 + m2 - a) * r02r12 + (m2 - b) * r12) % m2;
        ret[i] = a + b * w1 + u128(c) * w2;
    }
    return ret;
}
} // namespace ArbitraryNTT

template <typename mint> struct FormalPowerSeries : vector<mint> {
    using vector<mint>::vector;
    using FPS = FormalPowerSeries;

    FPS &operator+=(const FPS &r) {
        if(r.size() > this->size()) this->resize(r.size());
        for(int i = 0; i < (int)r.size(); i++) (*this)[i] += r[i];
        return *this;
    }

    FPS &operator+=(const mint &r) {
        if(this->empty()) this->resize(1);
        (*this)[0] += r;
        return *this;
    }

    FPS &operator-=(const FPS &r) {
        if(r.size() > this->size()) this->resize(r.size());
        for(int i = 0; i < (int)r.size(); i++) (*this)[i] -= r[i];
        return *this;
    }

    FPS &operator-=(const mint &r) {
        if(this->empty()) this->resize(1);
        (*this)[0] -= r;
        return *this;
    }

    FPS &operator*=(const mint &v) {
        for(int k = 0; k < (int)this->size(); k++) (*this)[k] *= v;
        return *this;
    }

    FPS &operator/=(const FPS &r) {
        if(this->size() < r.size()) {
            this->clear();
            return *this;
        }
        int n = this->size() - r.size() + 1;
        if((int)r.size() <= 64) {
            FPS f(*this), g(r);
            g.shrink();
            mint coeff = g.back().inverse();
            for(auto &x : g) x *= coeff;
            int deg = (int)f.size() - (int)g.size() + 1;
            int gs = g.size();
            FPS quo(deg);
            for(int i = deg - 1; i >= 0; i--) {
                quo[i] = f[i + gs - 1];
                for(int j = 0; j < gs; j++) f[i + j] -= quo[i] * g[j];
            }
            *this = quo * coeff;
            this->resize(n, mint(0));
            return *this;
        }
        return *this = ((*this).rev().pre(n) * r.rev().inv(n)).pre(n).rev();
    }

    FPS &operator%=(const FPS &r) {
        *this -= *this / r * r;
        shrink();
        return *this;
    }

    FPS operator+(const FPS &r) const { return FPS(*this) += r; }
    FPS operator+(const mint &v) const { return FPS(*this) += v; }
    FPS operator-(const FPS &r) const { return FPS(*this) -= r; }
    FPS operator-(const mint &v) const { return FPS(*this) -= v; }
    FPS operator*(const FPS &r) const { return FPS(*this) *= r; }
    FPS operator*(const mint &v) const { return FPS(*this) *= v; }
    FPS operator/(const FPS &r) const { return FPS(*this) /= r; }
    FPS operator%(const FPS &r) const { return FPS(*this) %= r; }
    FPS operator-() const {
        FPS ret(this->size());
        for(int i = 0; i < (int)this->size(); i++) ret[i] = -(*this)[i];
        return ret;
    }

    void shrink() {
        while(this->size() && this->back() == mint(0)) this->pop_back();
    }

    FPS rev() const {
        FPS ret(*this);
        reverse(begin(ret), end(ret));
        return ret;
    }

    FPS dot(FPS r) const {
        FPS ret(min(this->size(), r.size()));
        for(int i = 0; i < (int)ret.size(); i++) ret[i] = (*this)[i] * r[i];
        return ret;
    }

    // 前 sz 項を取ってくる。sz に足りない項は 0 埋めする
    FPS pre(int sz) const {
        FPS ret(begin(*this), begin(*this) + min((int)this->size(), sz));
        if((int)ret.size() < sz) ret.resize(sz);
        return ret;
    }

    FPS operator>>(int sz) const {
        if((int)this->size() <= sz) return {};
        FPS ret(*this);
        ret.erase(ret.begin(), ret.begin() + sz);
        return ret;
    }

    FPS operator<<(int sz) const {
        FPS ret(*this);
        ret.insert(ret.begin(), sz, mint(0));
        return ret;
    }

    FPS diff() const {
        const int n = (int)this->size();
        FPS ret(max(0, n - 1));
        mint one(1), coeff(1);
        for(int i = 1; i < n; i++) {
            ret[i - 1] = (*this)[i] * coeff;
            coeff += one;
        }
        return ret;
    }

    FPS integral() const {
        const int n = (int)this->size();
        FPS ret(n + 1);
        ret[0] = mint(0);
        if(n > 0) ret[1] = mint(1);
        auto mod = mint::get_mod();
        for(int i = 2; i <= n; i++) ret[i] = (-ret[mod % i]) * (mod / i);
        for(int i = 0; i < n; i++) ret[i + 1] *= (*this)[i];
        return ret;
    }

    mint eval(mint x) const {
        mint r = 0, w = 1;
        for(auto &v : *this) r += w * v, w *= x;
        return r;
    }

    FPS log(int deg = -1) const {
        assert(!(*this).empty() && (*this)[0] == mint(1));
        if(deg == -1) deg = (int)this->size();
        return (this->diff() * this->inv(deg)).pre(deg - 1).integral();
    }

    FPS pow(int64_t k, int deg = -1) const {
        const int n = (int)this->size();
        if(deg == -1) deg = n;
        if(k == 0) {
            FPS ret(deg);
            if(deg) ret[0] = 1;
            return ret;
        }
        for(int i = 0; i < n; i++) {
            if((*this)[i] != mint(0)) {
                mint rev = mint(1) / (*this)[i];
                FPS ret = (((*this * rev) >> i).log(deg) * k).exp(deg);
                ret *= (*this)[i].pow(k);
                ret = (ret << (i * k)).pre(deg);
                if((int)ret.size() < deg) ret.resize(deg, mint(0));
                return ret;
            }
            if(__int128_t(i + 1) * k >= deg) return FPS(deg, mint(0));
        }
        return FPS(deg, mint(0));
    }

    static void *ntt_ptr;
    static void set_fft();
    FPS &operator*=(const FPS &r);
    void ntt();
    void intt();
    void ntt_doubling();
    static int ntt_pr();
    FPS inv(int deg = -1) const;
    FPS exp(int deg = -1) const;
};
template <typename mint> void *FormalPowerSeries<mint>::ntt_ptr = nullptr;

/**
 * @brief 多項式/形式的冪級数ライブラリ
 * @docs docs/fps/formal-power-series.md
 */

template <typename mint> void FormalPowerSeries<mint>::set_fft() { ntt_ptr = nullptr; }

template <typename mint> void FormalPowerSeries<mint>::ntt() { exit(1); }

template <typename mint> void FormalPowerSeries<mint>::intt() { exit(1); }

template <typename mint> void FormalPowerSeries<mint>::ntt_doubling() { exit(1); }

template <typename mint> int FormalPowerSeries<mint>::ntt_pr() { exit(1); }

template <typename mint> FormalPowerSeries<mint> &FormalPowerSeries<mint>::operator*=(const FormalPowerSeries<mint> &r) {
    if(this->empty() || r.empty()) {
        this->clear();
        return *this;
    }
    auto ret = ArbitraryNTT::multiply(*this, r);
    return *this = FormalPowerSeries<mint>(ret.begin(), ret.end());
}

template <typename mint> FormalPowerSeries<mint> FormalPowerSeries<mint>::inv(int deg) const {
    assert((*this)[0] != mint(0));
    if(deg == -1) deg = (*this).size();
    FormalPowerSeries<mint> ret({mint(1) / (*this)[0]});
    for(int i = 1; i < deg; i <<= 1) ret = (ret + ret - ret * ret * (*this).pre(i << 1)).pre(i << 1);
    return ret.pre(deg);
}

template <typename mint> FormalPowerSeries<mint> FormalPowerSeries<mint>::exp(int deg) const {
    assert((*this).size() == 0 || (*this)[0] == mint(0));
    if(deg == -1) deg = (int)this->size();
    FormalPowerSeries<mint> ret({mint(1)});
    for(int i = 1; i < deg; i <<= 1) { ret = (ret * (pre(i << 1) + mint(1) - ret.log(i << 1))).pre(i << 1); }
    return ret.pre(deg);
}

struct Barrett {
    using u32 = unsigned int;
    using i64 = long long;
    using u64 = unsigned long long;
    u32 m;
    u64 im;
    Barrett() : m(), im() {}
    Barrett(int n) : m(n), im(u64(-1) / m + 1) {}
    constexpr inline i64 quo(u64 n) {
        u64 x = u64((__uint128_t(n) * im) >> 64);
        u32 r = n - x * m;
        return m <= r ? x - 1 : x;
    }
    constexpr inline i64 rem(u64 n) {
        u64 x = u64((__uint128_t(n) * im) >> 64);
        u32 r = n - x * m;
        return m <= r ? r + m : r;
    }
    constexpr inline pair<i64, int> quorem(u64 n) {
        u64 x = u64((__uint128_t(n) * im) >> 64);
        u32 r = n - x * m;
        if(m <= r) return {x - 1, r + m};
        return {x, r};
    }
    constexpr inline i64 pow(u64 n, i64 p) {
        u32 a = rem(n), r = m == 1 ? 0 : 1;
        while(p) {
            if(p & 1) r = rem(u64(r) * a);
            a = rem(u64(a) * a);
            p >>= 1;
        }
        return r;
    }
};

template <int id> struct ArbitraryModIntBase {
    int x;

    ArbitraryModIntBase() : x(0) {}

    ArbitraryModIntBase(int64_t y) {
        int z = y % get_mod();
        if(z < 0) z += get_mod();
        x = z;
    }

    ArbitraryModIntBase &operator+=(const ArbitraryModIntBase &p) {
        if((x += p.x) >= get_mod()) x -= get_mod();
        return *this;
    }

    ArbitraryModIntBase &operator-=(const ArbitraryModIntBase &p) {
        if((x += get_mod() - p.x) >= get_mod()) x -= get_mod();
        return *this;
    }

    ArbitraryModIntBase &operator*=(const ArbitraryModIntBase &p) {
        x = rem((unsigned long long)x * p.x);
        return *this;
    }

    ArbitraryModIntBase &operator/=(const ArbitraryModIntBase &p) {
        *this *= p.inverse();
        return *this;
    }

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

    ArbitraryModIntBase operator+(const ArbitraryModIntBase &p) const { return ArbitraryModIntBase(*this) += p; }

    ArbitraryModIntBase operator-(const ArbitraryModIntBase &p) const { return ArbitraryModIntBase(*this) -= p; }

    ArbitraryModIntBase operator*(const ArbitraryModIntBase &p) const { return ArbitraryModIntBase(*this) *= p; }

    ArbitraryModIntBase operator/(const ArbitraryModIntBase &p) const { return ArbitraryModIntBase(*this) /= p; }

    bool operator==(const ArbitraryModIntBase &p) const { return x == p.x; }

    bool operator!=(const ArbitraryModIntBase &p) const { return x != p.x; }

    ArbitraryModIntBase inverse() const {
        int a = x, b = get_mod(), u = 1, v = 0, t;
        while(b > 0) {
            t = a / b;
            swap(a -= t * b, b);
            swap(u -= t * v, v);
        }
        return ArbitraryModIntBase(u);
    }

    ArbitraryModIntBase pow(int64_t n) const {
        ArbitraryModIntBase ret(1), mul(x);
        while(n > 0) {
            if(n & 1) ret *= mul;
            mul *= mul;
            n >>= 1;
        }
        return ret;
    }

    friend ostream &operator<<(ostream &os, const ArbitraryModIntBase &p) { return os << p.x; }

    friend istream &operator>>(istream &is, ArbitraryModIntBase &a) {
        int64_t t;
        is >> t;
        a = ArbitraryModIntBase(t);
        return (is);
    }

    int get() const { return x; }

    inline unsigned int rem(unsigned long long p) { return barrett().rem(p); }

    static inline Barrett &barrett() {
        static Barrett b;
        return b;
    }

    static inline int &get_mod() {
        static int mod = 0;
        return mod;
    }

    static void set_mod(int md) {
        assert(0 < md && md <= (1LL << 30) - 1);
        get_mod() = md;
        barrett() = Barrett(md);
    }
};

using ArbitraryModInt = ArbitraryModIntBase<-1>;

using vmint = vector<ArbitraryModInt>;
using mint = ArbitraryModInt;
using fps = FormalPowerSeries<mint>;

constexpr int MAXN = 1e7;
vmint Fact, Ifact, Inv;
mint inv(int n) {
    if(n > MAXN) return (mint(n)).pow(mint::get_mod() - 2);
    if(Inv.empty()) Inv.emplace_back(0), Inv.emplace_back(1);
    if(Inv.size() > n)
        return Inv[n];
    else {
        for(int i = Inv.size(); i <= n; ++i) {
            auto [y, x] = div(int(mint::get_mod()), i);
            Inv.emplace_back(Inv[x] * (-y));
        }
        return Inv[n];
    }
}
mint fact(int n) {
    if(Fact.size() > n) return Fact[n];
    if(empty(Fact)) Fact = {1, 1};
    for(int i = Fact.size(); i <= n; ++i) Fact.emplace_back(Fact[i - 1] * i);
    return Fact[n];
}
mint ifact(int n) {
    if(Ifact.size() > n) return Ifact[n];
    if(empty(Ifact)) Ifact = {1, 1};
    for(int i = Ifact.size(); i <= n; ++i) Ifact.emplace_back(Ifact[i - 1] * inv(i));
    return Ifact[n];
}
mint modpow(ll a, ll n) { return mint(a).pow(n); }
mint inv(mint a) { return inv(a.get()); }
mint ifact(mint a) { return ifact(a.get()); }
mint fact(mint a) { return fact(a.get()); }
mint modpow(mint a, ll n) { return modpow(a.get(), n); }
mint C(int a, int b) {
    if(a < 0 || b < 0) return 0;
    if(a < b) return 0;
    if(a > MAXN) {
        mint res = 1;
        rep(i, b) res *= a - i, res /= i + 1;
        return res;
    }
    return fact(a) * ifact(b) * ifact(a - b);
}
mint P(int a, int b) {
    if(a < 0 || b < 0) return 0;
    if(a < b) return 0;
    if(a > MAXN) {
        mint res = 1;
        rep(i, b) res *= a - i;
        return res;
    }
    return fact(a) * ifact(a - b);
}

struct Prd {
    using Poly = fps;
    deque<Poly> deq;
    Prd() = default;
    void emplace(const Poly &f) { deq.emplace_back(f); }
    Poly calc() {
        if(deq.empty()) return {1};
        sort(all(deq), [&](const Poly &f, const Poly &g) { return si(f) < si(g); });
        while(deq.size() > 1) {
            deq.emplace_back(deq[0] * deq[1]);
            for(int i = 0; i < 2; ++i) deq.pop_front();
        }
        return deq.front();
    }
};

fps prd(vector<fps> &v) {
    Prd p;
    for(auto &e : v) p.emplace(e);
    return p.calc();
}

vmint power_table(mint x, int len) {
    vmint res(len + 1);
    res[0] = 1;
    rep(i, len) res[i + 1] = res[i] * x;
    return res;
}

// calc f(x + a)
fps TaylorShift(fps f, mint a) {
    int n = f.size();
    rep(i, n) f[i] *= fact(i);
    reverse(all(f));
    fps g(n, 1);
    rep(i, 1, n) g[i] = g[i - 1] * a * inv(i);
    f = (f * g);
    f.resize(n);
    reverse(begin(f), end(f));

    rep(i, n) f[i] *= ifact(i);
    return f;
}

// ボールの数、一個以上必要な数、入っていなくてもいい数(区別あり)
mint choose(int num, int a, int b = 0) {
    if(num == 0) return !a;
    return C(num + b - 1, a + b - 1);
}

// +1 n 個 -1 m 個で累積和 >= 0
mint Catalan(int n, int m) { return C(n + m, m) - C(n + m, m - 1); }

// +1 n 個 -1 m 個で累積和 > -k
mint Catalan2(int n, int m, int k) {
    if(m < k) return C(n + m, m);
    if(m < n + k) return C(n + m, m) - C(n + m, m - k);
    return 0;
}

// +1 n 個 -1 m 個で累積和 < +k
mint Catalan3(int n, int m, int k) { return Catalan2(m, n, k); }
string to_fraction(mint x) {
    static const int M = sqrtl(mint::get_mod());
    rep(i, 1, M + 1) {
        if((x * i).get() < M) return (i > 1 ? to_string((x * i).get()) + " / " + to_string(i) : to_string((x * i).get()));
        if(mint::get_mod() - (x * i).get() < M)
            return (i > 1 ? to_string(mint::get_mod() - (x * i).get()) + " / " + to_string(i) : to_string(mint::get_mod() - (x * i).get()));
    }
    return "?";
}
    constexpr int B = 2200;

mint dp[B  +1][B + 1];
int main() {
    INT(n, m);
    ArbitraryModInt::set_mod(m);
    n += 10;

auto ps = power_table(-2, n + 1);

    rep(j, B + 1) {
        rep(k, j + 1) {
                    dp[j][k] = C(j, k) * ps[k];
                  
                }
    }
    

   
    auto ps2 = power_table(2, n + 1);

    fps res;

    int mn = (n + 1) / 2;
    for(int l = 1; l <= mn;) {
        fps now(n + 1);
        int r = min(l + B, mn + 1);
        if(l == 1) {
            rep(i, l, r) {
                per(j, n + 1, 1) now[j] -= now[j - 1] + now[j - 1];
                rep(s, 2 * i, n + 1, i) now[s] += (i & 1 ? 1 : -1);
            }
        } else {
            rep(i, l, r) {
                int j = r - 1 - i;
                if(~i&1) rep(k, j+1) dp[j][k] = -dp[j][k];
                rep(s, 2 * i, n + 1, i) {
                    int m = min(j, n - s); 
                    rep(k, m + 1) {
                    now[s + k] += dp[j][k]; }
                    }

if(~i&1) rep(k, j+1) dp[j][k] = -dp[j][k];

            }
        }
        int t = n - (r - 1);
        fps h(t + 1);
        rep(i, t + 1) h[i] = C(t, i) * ps[i];

        res += ((now >> l * 2) * h) << l * 2;
        l = r;
    }
    fps g(n + 1);
    rep(i, n + 1) g[i] = C(i + n - 1, n - 1) * ps2[i];
    res *= g;
    res = fps(rng(res, 2, 2 + n - 10));
    OUT(res);
}