#include <immintrin.h>
#include <bits/stdc++.h>
using namespace std;

template <class T> int sz(T&& a) { return int(size(forward<T>(a))); }

template <class T> using vc = vector<T>;
template <class T> using vvc = vc<vc<T>>;

using ll = int64_t;
using vi = vc<int>;
using pii = pair<int, int>;
using uint = uint32_t;
using ull = uint64_t;

mt19937 mt(chrono::steady_clock::now().time_since_epoch().count());
int rand_int(int a, int b) {
	return uniform_int_distribution<int>(a, b)(mt);
}

template <class F> struct ycr {
	F f;
	template <class T> explicit ycr(T&& f_) : f(forward<T>(f_)) {}
	template <class... A> decltype(auto) operator()(A&&... args) {
		return f(ref(*this), forward<A>(args)...);
	}
};
template <class F> decltype(auto) yc(F&& f) {
	return ycr<decay_t<F>>(forward<F>(f));
}

#ifdef LOCAL
#define dbg(x) cerr << __LINE__ << " : " << #x << " = " << x << endl;
#else
#define dbg(x) true
#endif

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(r * mod == 1, "invalid, r * mod != 1");
  static_assert(mod < (1 << 30), "invalid, mod >= 2 ^ 30");
  static_assert((mod & 1) == 1, "invalid, mod % 2 == 0");

  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 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 { return pow(mod - 2); }

  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; }
};
//




#include <immintrin.h>

__attribute__((target("sse4.2"))) __attribute__((always_inline)) __m128i
my128_mullo_epu32(const __m128i &a, const __m128i &b) {
  return _mm_mullo_epi32(a, b);
}

__attribute__((target("sse4.2"))) __attribute__((always_inline)) __m128i
my128_mulhi_epu32(const __m128i &a, const __m128i &b) {
  __m128i a13 = _mm_shuffle_epi32(a, 0xF5);
  __m128i b13 = _mm_shuffle_epi32(b, 0xF5);
  __m128i prod02 = _mm_mul_epu32(a, b);
  __m128i prod13 = _mm_mul_epu32(a13, b13);
  __m128i prod = _mm_unpackhi_epi64(_mm_unpacklo_epi32(prod02, prod13),
                                    _mm_unpackhi_epi32(prod02, prod13));
  return prod;
}

__attribute__((target("sse4.2"))) __attribute__((always_inline)) __m128i
montgomery_mul_128(const __m128i &a, const __m128i &b, const __m128i &r,
                   const __m128i &m1) {
  return _mm_sub_epi32(
      _mm_add_epi32(my128_mulhi_epu32(a, b), m1),
      my128_mulhi_epu32(my128_mullo_epu32(my128_mullo_epu32(a, b), r), m1));
}

__attribute__((target("sse4.2"))) __attribute__((always_inline)) __m128i
montgomery_add_128(const __m128i &a, const __m128i &b, const __m128i &m2,
                   const __m128i &m0) {
  __m128i ret = _mm_sub_epi32(_mm_add_epi32(a, b), m2);
  return _mm_add_epi32(_mm_and_si128(_mm_cmpgt_epi32(m0, ret), m2), ret);
}

__attribute__((target("sse4.2"))) __attribute__((always_inline)) __m128i
montgomery_sub_128(const __m128i &a, const __m128i &b, const __m128i &m2,
                   const __m128i &m0) {
  __m128i ret = _mm_sub_epi32(a, b);
  return _mm_add_epi32(_mm_and_si128(_mm_cmpgt_epi32(m0, ret), m2), ret);
}

__attribute__((target("avx2"))) __attribute__((always_inline)) __m256i
my256_mullo_epu32(const __m256i &a, const __m256i &b) {
  return _mm256_mullo_epi32(a, b);
}

__attribute__((target("avx2"))) __attribute__((always_inline)) __m256i
my256_mulhi_epu32(const __m256i &a, const __m256i &b) {
  __m256i a13 = _mm256_shuffle_epi32(a, 0xF5);
  __m256i b13 = _mm256_shuffle_epi32(b, 0xF5);
  __m256i prod02 = _mm256_mul_epu32(a, b);
  __m256i prod13 = _mm256_mul_epu32(a13, b13);
  __m256i prod = _mm256_unpackhi_epi64(_mm256_unpacklo_epi32(prod02, prod13),
                                       _mm256_unpackhi_epi32(prod02, prod13));
  return prod;
}

__attribute__((target("avx2"))) __attribute__((always_inline)) __m256i
montgomery_mul_256(const __m256i &a, const __m256i &b, const __m256i &r,
                   const __m256i &m1) {
  return _mm256_sub_epi32(
      _mm256_add_epi32(my256_mulhi_epu32(a, b), m1),
      my256_mulhi_epu32(my256_mullo_epu32(my256_mullo_epu32(a, b), r), m1));
}

__attribute__((target("avx2"))) __attribute__((always_inline)) __m256i
montgomery_add_256(const __m256i &a, const __m256i &b, const __m256i &m2,
                   const __m256i &m0) {
  __m256i ret = _mm256_sub_epi32(_mm256_add_epi32(a, b), m2);
  return _mm256_add_epi32(_mm256_and_si256(_mm256_cmpgt_epi32(m0, ret), m2),
                          ret);
}

__attribute__((target("avx2"))) __attribute__((always_inline)) __m256i
montgomery_sub_256(const __m256i &a, const __m256i &b, const __m256i &m2,
                   const __m256i &m0) {
  __m256i ret = _mm256_sub_epi32(a, b);
  return _mm256_add_epi32(_mm256_and_si256(_mm256_cmpgt_epi32(m0, ret), m2),
                          ret);
}
namespace ntt_inner {
using u64 = uint64_t;
constexpr uint32_t get_pr(uint32_t mod) {
  if (mod == 2) return 1;
  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;
}

constexpr int SZ_FFT_BUF = 1 << 23;
uint32_t _buf1[SZ_FFT_BUF] __attribute__((aligned(64)));
uint32_t _buf2[SZ_FFT_BUF] __attribute__((aligned(64)));
}  // namespace ntt_inner

template <typename mint>
struct NTT {
  static constexpr uint32_t mod = mint::get_mod();
  static constexpr uint32_t pr = ntt_inner::get_pr(mint::get_mod());
  static constexpr int level = __builtin_ctzll(mod - 1);
  mint dw[level], dy[level];
  mint *buf1, *buf2;

  constexpr NTT() {
    setwy(level);
    union raw_cast {
      mint dat;
      uint32_t _;
    };
    buf1 = &(((raw_cast *)(ntt_inner::_buf1))->dat);
    buf2 = &(((raw_cast *)(ntt_inner::_buf2))->dat);
  }

  constexpr 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[0] = dy[0] = w[1] * w[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];
    }
  }

  __attribute__((target("avx2"))) void ntt(mint *a, int n) {
    int k = n ? __builtin_ctz(n) : 0;
    if (k == 0) 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);
      if (v < 8) {
        for (int j = 0; j < v; ++j) {
          mint ajv = a[j + v];
          a[j + v] = a[j] - ajv;
          a[j] += ajv;
        }
      } else {
        const __m256i m0 = _mm256_set1_epi32(0);
        const __m256i m2 = _mm256_set1_epi32(mod + mod);
        int j0 = 0;
        int j1 = v;
        for (; j0 < v; j0 += 8, j1 += 8) {
          __m256i T0 = _mm256_loadu_si256((__m256i *)(a + j0));
          __m256i T1 = _mm256_loadu_si256((__m256i *)(a + j1));
          __m256i naj = montgomery_add_256(T0, T1, m2, m0);
          __m256i najv = montgomery_sub_256(T0, T1, m2, m0);
          _mm256_storeu_si256((__m256i *)(a + j0), naj);
          _mm256_storeu_si256((__m256i *)(a + j1), najv);
        }
      }
    }
    int u = 1 << (2 + (k & 1));
    int v = 1 << (k - 2 - (k & 1));
    mint one = mint(1);
    mint imag = dw[1];
    while (v) {
      if (v == 1) {
        mint ww = one, xx = one, wx = one;
        for (int jh = 0; jh < u;) {
          ww = xx * xx, wx = ww * xx;
          mint t0 = a[jh + 0], t1 = a[jh + 1] * xx;
          mint t2 = a[jh + 2] * ww, t3 = a[jh + 3] * wx;
          mint t0p2 = t0 + t2, t1p3 = t1 + t3;
          mint t0m2 = t0 - t2, t1m3 = (t1 - t3) * imag;
          a[jh + 0] = t0p2 + t1p3, a[jh + 1] = t0p2 - t1p3;
          a[jh + 2] = t0m2 + t1m3, a[jh + 3] = t0m2 - t1m3;
          xx *= dw[__builtin_ctz((jh += 4))];
        }
      } else if (v == 4) {
        const __m128i m0 = _mm_set1_epi32(0);
        const __m128i m1 = _mm_set1_epi32(mod);
        const __m128i m2 = _mm_set1_epi32(mod + mod);
        const __m128i r = _mm_set1_epi32(mint::r);
        const __m128i Imag = _mm_set1_epi32(imag.a);
        mint ww = one, xx = one, wx = one;
        for (int jh = 0; jh < u;) {
          if (jh == 0) {
            int j0 = 0;
            int j1 = v;
            int j2 = j1 + v;
            int j3 = j2 + v;
            int je = v;
            for (; j0 < je; j0 += 4, j1 += 4, j2 += 4, j3 += 4) {
              const __m128i T0 = _mm_loadu_si128((__m128i *)(a + j0));
              const __m128i T1 = _mm_loadu_si128((__m128i *)(a + j1));
              const __m128i T2 = _mm_loadu_si128((__m128i *)(a + j2));
              const __m128i T3 = _mm_loadu_si128((__m128i *)(a + j3));
              const __m128i T0P2 = montgomery_add_128(T0, T2, m2, m0);
              const __m128i T1P3 = montgomery_add_128(T1, T3, m2, m0);
              const __m128i T0M2 = montgomery_sub_128(T0, T2, m2, m0);
              const __m128i T1M3 = montgomery_mul_128(
                  montgomery_sub_128(T1, T3, m2, m0), Imag, r, m1);
              _mm_storeu_si128((__m128i *)(a + j0),
                               montgomery_add_128(T0P2, T1P3, m2, m0));
              _mm_storeu_si128((__m128i *)(a + j1),
                               montgomery_sub_128(T0P2, T1P3, m2, m0));
              _mm_storeu_si128((__m128i *)(a + j2),
                               montgomery_add_128(T0M2, T1M3, m2, m0));
              _mm_storeu_si128((__m128i *)(a + j3),
                               montgomery_sub_128(T0M2, T1M3, m2, m0));
            }
          } else {
            ww = xx * xx, wx = ww * xx;
            const __m128i WW = _mm_set1_epi32(ww.a);
            const __m128i WX = _mm_set1_epi32(wx.a);
            const __m128i XX = _mm_set1_epi32(xx.a);
            int j0 = jh * v;
            int j1 = j0 + v;
            int j2 = j1 + v;
            int j3 = j2 + v;
            int je = j1;
            for (; j0 < je; j0 += 4, j1 += 4, j2 += 4, j3 += 4) {
              const __m128i T0 = _mm_loadu_si128((__m128i *)(a + j0));
              const __m128i T1 = _mm_loadu_si128((__m128i *)(a + j1));
              const __m128i T2 = _mm_loadu_si128((__m128i *)(a + j2));
              const __m128i T3 = _mm_loadu_si128((__m128i *)(a + j3));
              const __m128i MT1 = montgomery_mul_128(T1, XX, r, m1);
              const __m128i MT2 = montgomery_mul_128(T2, WW, r, m1);
              const __m128i MT3 = montgomery_mul_128(T3, WX, r, m1);
              const __m128i T0P2 = montgomery_add_128(T0, MT2, m2, m0);
              const __m128i T1P3 = montgomery_add_128(MT1, MT3, m2, m0);
              const __m128i T0M2 = montgomery_sub_128(T0, MT2, m2, m0);
              const __m128i T1M3 = montgomery_mul_128(
                  montgomery_sub_128(MT1, MT3, m2, m0), Imag, r, m1);
              _mm_storeu_si128((__m128i *)(a + j0),
                               montgomery_add_128(T0P2, T1P3, m2, m0));
              _mm_storeu_si128((__m128i *)(a + j1),
                               montgomery_sub_128(T0P2, T1P3, m2, m0));
              _mm_storeu_si128((__m128i *)(a + j2),
                               montgomery_add_128(T0M2, T1M3, m2, m0));
              _mm_storeu_si128((__m128i *)(a + j3),
                               montgomery_sub_128(T0M2, T1M3, m2, m0));
            }
          }
          xx *= dw[__builtin_ctz((jh += 4))];
        }
      } else {
        const __m256i m0 = _mm256_set1_epi32(0);
        const __m256i m1 = _mm256_set1_epi32(mod);
        const __m256i m2 = _mm256_set1_epi32(mod + mod);
        const __m256i r = _mm256_set1_epi32(mint::r);
        const __m256i Imag = _mm256_set1_epi32(imag.a);
        mint ww = one, xx = one, wx = one;
        for (int jh = 0; jh < u;) {
          if (jh == 0) {
            int j0 = 0;
            int j1 = v;
            int j2 = j1 + v;
            int j3 = j2 + v;
            int je = v;
            for (; j0 < je; j0 += 8, j1 += 8, j2 += 8, j3 += 8) {
              const __m256i T0 = _mm256_loadu_si256((__m256i *)(a + j0));
              const __m256i T1 = _mm256_loadu_si256((__m256i *)(a + j1));
              const __m256i T2 = _mm256_loadu_si256((__m256i *)(a + j2));
              const __m256i T3 = _mm256_loadu_si256((__m256i *)(a + j3));
              const __m256i T0P2 = montgomery_add_256(T0, T2, m2, m0);
              const __m256i T1P3 = montgomery_add_256(T1, T3, m2, m0);
              const __m256i T0M2 = montgomery_sub_256(T0, T2, m2, m0);
              const __m256i T1M3 = montgomery_mul_256(
                  montgomery_sub_256(T1, T3, m2, m0), Imag, r, m1);
              _mm256_storeu_si256((__m256i *)(a + j0),
                                  montgomery_add_256(T0P2, T1P3, m2, m0));
              _mm256_storeu_si256((__m256i *)(a + j1),
                                  montgomery_sub_256(T0P2, T1P3, m2, m0));
              _mm256_storeu_si256((__m256i *)(a + j2),
                                  montgomery_add_256(T0M2, T1M3, m2, m0));
              _mm256_storeu_si256((__m256i *)(a + j3),
                                  montgomery_sub_256(T0M2, T1M3, m2, m0));
            }
          } else {
            ww = xx * xx, wx = ww * xx;
            const __m256i WW = _mm256_set1_epi32(ww.a);
            const __m256i WX = _mm256_set1_epi32(wx.a);
            const __m256i XX = _mm256_set1_epi32(xx.a);
            int j0 = jh * v;
            int j1 = j0 + v;
            int j2 = j1 + v;
            int j3 = j2 + v;
            int je = j1;
            for (; j0 < je; j0 += 8, j1 += 8, j2 += 8, j3 += 8) {
              const __m256i T0 = _mm256_loadu_si256((__m256i *)(a + j0));
              const __m256i T1 = _mm256_loadu_si256((__m256i *)(a + j1));
              const __m256i T2 = _mm256_loadu_si256((__m256i *)(a + j2));
              const __m256i T3 = _mm256_loadu_si256((__m256i *)(a + j3));
              const __m256i MT1 = montgomery_mul_256(T1, XX, r, m1);
              const __m256i MT2 = montgomery_mul_256(T2, WW, r, m1);
              const __m256i MT3 = montgomery_mul_256(T3, WX, r, m1);
              const __m256i T0P2 = montgomery_add_256(T0, MT2, m2, m0);
              const __m256i T1P3 = montgomery_add_256(MT1, MT3, m2, m0);
              const __m256i T0M2 = montgomery_sub_256(T0, MT2, m2, m0);
              const __m256i T1M3 = montgomery_mul_256(
                  montgomery_sub_256(MT1, MT3, m2, m0), Imag, r, m1);
              _mm256_storeu_si256((__m256i *)(a + j0),
                                  montgomery_add_256(T0P2, T1P3, m2, m0));
              _mm256_storeu_si256((__m256i *)(a + j1),
                                  montgomery_sub_256(T0P2, T1P3, m2, m0));
              _mm256_storeu_si256((__m256i *)(a + j2),
                                  montgomery_add_256(T0M2, T1M3, m2, m0));
              _mm256_storeu_si256((__m256i *)(a + j3),
                                  montgomery_sub_256(T0M2, T1M3, m2, m0));
            }
          }
          xx *= dw[__builtin_ctz((jh += 4))];
        }
      }
      u <<= 2;
      v >>= 2;
    }
  }

  __attribute__((target("avx2"))) void intt(mint *a, int n,
                                            int normalize = true) {
    int k = n ? __builtin_ctz(n) : 0;
    if (k == 0) return;
    if (k == 1) {
      mint a1 = a[1];
      a[1] = a[0] - a[1];
      a[0] = a[0] + a1;
      if (normalize) {
        a[0] *= mint(2).inverse();
        a[1] *= mint(2).inverse();
      }
      return;
    }
    int u = 1 << (k - 2);
    int v = 1;
    mint one = mint(1);
    mint imag = dy[1];
    while (u) {
      if (v == 1) {
        mint ww = one, xx = one, yy = one;
        u <<= 2;
        for (int jh = 0; jh < u;) {
          ww = xx * xx, yy = xx * imag;
          mint t0 = a[jh + 0], t1 = a[jh + 1];
          mint t2 = a[jh + 2], t3 = a[jh + 3];
          mint t0p1 = t0 + t1, t2p3 = t2 + t3;
          mint t0m1 = (t0 - t1) * xx, t2m3 = (t2 - t3) * yy;
          a[jh + 0] = t0p1 + t2p3, a[jh + 2] = (t0p1 - t2p3) * ww;
          a[jh + 1] = t0m1 + t2m3, a[jh + 3] = (t0m1 - t2m3) * ww;
          xx *= dy[__builtin_ctz(jh += 4)];
        }
      } else if (v == 4) {
        const __m128i m0 = _mm_set1_epi32(0);
        const __m128i m1 = _mm_set1_epi32(mod);
        const __m128i m2 = _mm_set1_epi32(mod + mod);
        const __m128i r = _mm_set1_epi32(mint::r);
        const __m128i Imag = _mm_set1_epi32(imag.a);
        mint ww = one, xx = one, yy = one;
        u <<= 2;
        for (int jh = 0; jh < u;) {
          if (jh == 0) {
            int j0 = 0;
            int j1 = v;
            int j2 = v + v;
            int j3 = j2 + v;
            for (; j0 < v; j0 += 4, j1 += 4, j2 += 4, j3 += 4) {
              const __m128i T0 = _mm_loadu_si128((__m128i *)(a + j0));
              const __m128i T1 = _mm_loadu_si128((__m128i *)(a + j1));
              const __m128i T2 = _mm_loadu_si128((__m128i *)(a + j2));
              const __m128i T3 = _mm_loadu_si128((__m128i *)(a + j3));
              const __m128i T0P1 = montgomery_add_128(T0, T1, m2, m0);
              const __m128i T2P3 = montgomery_add_128(T2, T3, m2, m0);
              const __m128i T0M1 = montgomery_sub_128(T0, T1, m2, m0);
              const __m128i T2M3 = montgomery_mul_128(
                  montgomery_sub_128(T2, T3, m2, m0), Imag, r, m1);
              _mm_storeu_si128((__m128i *)(a + j0),
                               montgomery_add_128(T0P1, T2P3, m2, m0));
              _mm_storeu_si128((__m128i *)(a + j2),
                               montgomery_sub_128(T0P1, T2P3, m2, m0));
              _mm_storeu_si128((__m128i *)(a + j1),
                               montgomery_add_128(T0M1, T2M3, m2, m0));
              _mm_storeu_si128((__m128i *)(a + j3),
                               montgomery_sub_128(T0M1, T2M3, m2, m0));
            }
          } else {
            ww = xx * xx, yy = xx * imag;
            const __m128i WW = _mm_set1_epi32(ww.a);
            const __m128i XX = _mm_set1_epi32(xx.a);
            const __m128i YY = _mm_set1_epi32(yy.a);
            int j0 = jh * v;
            int j1 = j0 + v;
            int j2 = j1 + v;
            int j3 = j2 + v;
            int je = j1;
            for (; j0 < je; j0 += 4, j1 += 4, j2 += 4, j3 += 4) {
              const __m128i T0 = _mm_loadu_si128((__m128i *)(a + j0));
              const __m128i T1 = _mm_loadu_si128((__m128i *)(a + j1));
              const __m128i T2 = _mm_loadu_si128((__m128i *)(a + j2));
              const __m128i T3 = _mm_loadu_si128((__m128i *)(a + j3));
              const __m128i T0P1 = montgomery_add_128(T0, T1, m2, m0);
              const __m128i T2P3 = montgomery_add_128(T2, T3, m2, m0);
              const __m128i T0M1 = montgomery_mul_128(
                  montgomery_sub_128(T0, T1, m2, m0), XX, r, m1);
              __m128i T2M3 = montgomery_mul_128(
                  montgomery_sub_128(T2, T3, m2, m0), YY, r, m1);
              _mm_storeu_si128((__m128i *)(a + j0),
                               montgomery_add_128(T0P1, T2P3, m2, m0));
              _mm_storeu_si128(
                  (__m128i *)(a + j2),
                  montgomery_mul_128(montgomery_sub_128(T0P1, T2P3, m2, m0), WW,
                                     r, m1));
              _mm_storeu_si128((__m128i *)(a + j1),
                               montgomery_add_128(T0M1, T2M3, m2, m0));
              _mm_storeu_si128(
                  (__m128i *)(a + j3),
                  montgomery_mul_128(montgomery_sub_128(T0M1, T2M3, m2, m0), WW,
                                     r, m1));
            }
          }
          xx *= dy[__builtin_ctz(jh += 4)];
        }
      } else {
        const __m256i m0 = _mm256_set1_epi32(0);
        const __m256i m1 = _mm256_set1_epi32(mod);
        const __m256i m2 = _mm256_set1_epi32(mod + mod);
        const __m256i r = _mm256_set1_epi32(mint::r);
        const __m256i Imag = _mm256_set1_epi32(imag.a);
        mint ww = one, xx = one, yy = one;
        u <<= 2;
        for (int jh = 0; jh < u;) {
          if (jh == 0) {
            int j0 = 0;
            int j1 = v;
            int j2 = v + v;
            int j3 = j2 + v;
            for (; j0 < v; j0 += 8, j1 += 8, j2 += 8, j3 += 8) {
              const __m256i T0 = _mm256_loadu_si256((__m256i *)(a + j0));
              const __m256i T1 = _mm256_loadu_si256((__m256i *)(a + j1));
              const __m256i T2 = _mm256_loadu_si256((__m256i *)(a + j2));
              const __m256i T3 = _mm256_loadu_si256((__m256i *)(a + j3));
              const __m256i T0P1 = montgomery_add_256(T0, T1, m2, m0);
              const __m256i T2P3 = montgomery_add_256(T2, T3, m2, m0);
              const __m256i T0M1 = montgomery_sub_256(T0, T1, m2, m0);
              const __m256i T2M3 = montgomery_mul_256(
                  montgomery_sub_256(T2, T3, m2, m0), Imag, r, m1);
              _mm256_storeu_si256((__m256i *)(a + j0),
                                  montgomery_add_256(T0P1, T2P3, m2, m0));
              _mm256_storeu_si256((__m256i *)(a + j2),
                                  montgomery_sub_256(T0P1, T2P3, m2, m0));
              _mm256_storeu_si256((__m256i *)(a + j1),
                                  montgomery_add_256(T0M1, T2M3, m2, m0));
              _mm256_storeu_si256((__m256i *)(a + j3),
                                  montgomery_sub_256(T0M1, T2M3, m2, m0));
            }
          } else {
            ww = xx * xx, yy = xx * imag;
            const __m256i WW = _mm256_set1_epi32(ww.a);
            const __m256i XX = _mm256_set1_epi32(xx.a);
            const __m256i YY = _mm256_set1_epi32(yy.a);
            int j0 = jh * v;
            int j1 = j0 + v;
            int j2 = j1 + v;
            int j3 = j2 + v;
            int je = j1;
            for (; j0 < je; j0 += 8, j1 += 8, j2 += 8, j3 += 8) {
              const __m256i T0 = _mm256_loadu_si256((__m256i *)(a + j0));
              const __m256i T1 = _mm256_loadu_si256((__m256i *)(a + j1));
              const __m256i T2 = _mm256_loadu_si256((__m256i *)(a + j2));
              const __m256i T3 = _mm256_loadu_si256((__m256i *)(a + j3));
              const __m256i T0P1 = montgomery_add_256(T0, T1, m2, m0);
              const __m256i T2P3 = montgomery_add_256(T2, T3, m2, m0);
              const __m256i T0M1 = montgomery_mul_256(
                  montgomery_sub_256(T0, T1, m2, m0), XX, r, m1);
              const __m256i T2M3 = montgomery_mul_256(
                  montgomery_sub_256(T2, T3, m2, m0), YY, r, m1);
              _mm256_storeu_si256((__m256i *)(a + j0),
                                  montgomery_add_256(T0P1, T2P3, m2, m0));
              _mm256_storeu_si256(
                  (__m256i *)(a + j2),
                  montgomery_mul_256(montgomery_sub_256(T0P1, T2P3, m2, m0), WW,
                                     r, m1));
              _mm256_storeu_si256((__m256i *)(a + j1),
                                  montgomery_add_256(T0M1, T2M3, m2, m0));
              _mm256_storeu_si256(
                  (__m256i *)(a + j3),
                  montgomery_mul_256(montgomery_sub_256(T0M1, T2M3, m2, m0), WW,
                                     r, m1));
            }
          }
          xx *= dy[__builtin_ctz(jh += 4)];
        }
      }
      u >>= 4;
      v <<= 2;
    }
    if (k & 1) {
      v = 1 << (k - 1);
      if (v < 8) {
        for (int j = 0; j < v; ++j) {
          mint ajv = a[j] - a[j + v];
          a[j] += a[j + v];
          a[j + v] = ajv;
        }
      } else {
        const __m256i m0 = _mm256_set1_epi32(0);
        const __m256i m2 = _mm256_set1_epi32(mod + mod);
        int j0 = 0;
        int j1 = v;
        for (; j0 < v; j0 += 8, j1 += 8) {
          const __m256i T0 = _mm256_loadu_si256((__m256i *)(a + j0));
          const __m256i T1 = _mm256_loadu_si256((__m256i *)(a + j1));
          __m256i naj = montgomery_add_256(T0, T1, m2, m0);
          __m256i najv = montgomery_sub_256(T0, T1, m2, m0);
          _mm256_storeu_si256((__m256i *)(a + j0), naj);
          _mm256_storeu_si256((__m256i *)(a + j1), najv);
        }
      }
    }
    if (normalize) {
      mint invn = mint(n).inverse();
      for (int i = 0; i < n; i++) a[i] *= invn;
    }
  }

  __attribute__((target("avx2"))) void inplace_multiply(
      int l1, int l2, int zero_padding = true) {
    int l = l1 + l2 - 1;
    int M = 4;
    while (M < l) M <<= 1;
    if (zero_padding) {
      for (int i = l1; i < M; i++) ntt_inner::_buf1[i] = 0;
      for (int i = l2; i < M; i++) ntt_inner::_buf2[i] = 0;
    }
    const __m256i m0 = _mm256_set1_epi32(0);
    const __m256i m1 = _mm256_set1_epi32(mod);
    const __m256i r = _mm256_set1_epi32(mint::r);
    const __m256i N2 = _mm256_set1_epi32(mint::n2);
    for (int i = 0; i < l1; i += 8) {
      __m256i a = _mm256_loadu_si256((__m256i *)(ntt_inner::_buf1 + i));
      __m256i b = montgomery_mul_256(a, N2, r, m1);
      _mm256_storeu_si256((__m256i *)(ntt_inner::_buf1 + i), b);
    }
    for (int i = 0; i < l2; i += 8) {
      __m256i a = _mm256_loadu_si256((__m256i *)(ntt_inner::_buf2 + i));
      __m256i b = montgomery_mul_256(a, N2, r, m1);
      _mm256_storeu_si256((__m256i *)(ntt_inner::_buf2 + i), b);
    }
    ntt(buf1, M);
    ntt(buf2, M);
    for (int i = 0; i < M; i += 8) {
      __m256i a = _mm256_loadu_si256((__m256i *)(ntt_inner::_buf1 + i));
      __m256i b = _mm256_loadu_si256((__m256i *)(ntt_inner::_buf2 + i));
      __m256i c = montgomery_mul_256(a, b, r, m1);
      _mm256_storeu_si256((__m256i *)(ntt_inner::_buf1 + i), c);
    }
    intt(buf1, M, false);
    const __m256i INVM = _mm256_set1_epi32((mint(M).inverse()).a);
    for (int i = 0; i < l; i += 8) {
      __m256i a = _mm256_loadu_si256((__m256i *)(ntt_inner::_buf1 + i));
      __m256i b = montgomery_mul_256(a, INVM, r, m1);
      __m256i c = my256_mulhi_epu32(my256_mullo_epu32(b, r), m1);
      __m256i d = _mm256_and_si256(_mm256_cmpgt_epi32(c, m0), m1);
      __m256i e = _mm256_sub_epi32(d, c);
      _mm256_storeu_si256((__m256i *)(ntt_inner::_buf1 + i), e);
    }
  }

  void ntt(vector<mint> &a) {
    int M = (int)a.size();
    for (int i = 0; i < M; i++) buf1[i].a = a[i].a;
    ntt(buf1, M);
    for (int i = 0; i < M; i++) a[i].a = buf1[i].a;
  }

  void intt(vector<mint> &a) {
    int M = (int)a.size();
    for (int i = 0; i < M; i++) buf1[i].a = a[i].a;
    intt(buf1, M, true);
    for (int i = 0; i < M; i++) a[i].a = buf1[i].a;
  }

  vector<mint> multiply(const vector<mint> &a, const vector<mint> &b) {
    if (a.size() == 0 && b.size() == 0) return vector<mint>{};
    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;
    }
    assert(l <= ntt_inner::SZ_FFT_BUF);
    int M = 4;
    while (M < l) M <<= 1;
    for (int i = 0; i < (int)a.size(); ++i) buf1[i].a = a[i].a;
    for (int i = (int)a.size(); i < M; ++i) buf1[i].a = 0;
    for (int i = 0; i < (int)b.size(); ++i) buf2[i].a = b[i].a;
    for (int i = (int)b.size(); i < M; ++i) buf2[i].a = 0;
    ntt(buf1, M);
    ntt(buf2, M);
    for (int i = 0; i < M; ++i)
      buf1[i].a = mint::reduce(uint64_t(buf1[i].a) * buf2[i].a);
    intt(buf1, M, false);
    vector<mint> s(l);
    mint invm = mint(M).inverse();
    for (int i = 0; i < l; ++i) s[i] = buf1[i] * invm;
    return s;
  }

  void ntt_doubling(vector<mint> &a) {
    int M = (int)a.size();
    for (int i = 0; i < M; i++) buf1[i].a = a[i].a;
    intt(buf1, M);
    mint r = 1, zeta = mint(pr).pow((mint::get_mod() - 1) / (M << 1));
    for (int i = 0; i < M; i++) buf1[i] *= r, r *= zeta;
    ntt(buf1, M);
    a.resize(2 * M);
    for (int i = 0; i < M; i++) a[M + i].a = buf1[i].a;
  }
};

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;
  }

  FPS pre(int sz) const {
    return FPS(begin(*this), begin(*this) + min((int)this->size(), sz));
  }

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

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

template <typename mint>
void FormalPowerSeries<mint>::ntt() {
  set_fft();
  static_cast<NTT<mint>*>(ntt_ptr)->ntt(*this);
}

template <typename mint>
void FormalPowerSeries<mint>::intt() {
  set_fft();
  static_cast<NTT<mint>*>(ntt_ptr)->intt(*this);
}

template <typename mint>
void FormalPowerSeries<mint>::ntt_doubling() {
  set_fft();
  static_cast<NTT<mint>*>(ntt_ptr)->ntt_doubling(*this);
}

template <typename mint>
int FormalPowerSeries<mint>::ntt_pr() {
  set_fft();
  return static_cast<NTT<mint>*>(ntt_ptr)->pr;
}

template <typename mint>
FormalPowerSeries<mint> FormalPowerSeries<mint>::inv(int deg) const {
  assert((*this)[0] != mint(0));
  if (deg == -1) deg = (int)this->size();
  FormalPowerSeries<mint> res(deg);
  res[0] = {mint(1) / (*this)[0]};
  for (int d = 1; d < deg; d <<= 1) {
    FormalPowerSeries<mint> f(2 * d), g(2 * d);
    for (int j = 0; j < min((int)this->size(), 2 * d); j++) f[j] = (*this)[j];
    for (int j = 0; j < d; j++) g[j] = res[j];
    f.ntt();
    g.ntt();
    for (int j = 0; j < 2 * d; j++) f[j] *= g[j];
    f.intt();
    for (int j = 0; j < d; j++) f[j] = 0;
    f.ntt();
    for (int j = 0; j < 2 * d; j++) f[j] *= g[j];
    f.intt();
    for (int j = d; j < min(2 * d, deg); j++) res[j] = -f[j];
  }
  return res.pre(deg);
}

template <typename mint>
FormalPowerSeries<mint> FormalPowerSeries<mint>::exp(int deg) const {
  using fps = FormalPowerSeries<mint>;
  assert((*this).size() == 0 || (*this)[0] == mint(0));
  if (deg == -1) deg = this->size();

  fps inv;
  inv.reserve(deg + 1);
  inv.push_back(mint(0));
  inv.push_back(mint(1));

  auto inplace_integral = [&](fps& F) -> void {
    const int n = (int)F.size();
    auto mod = mint::get_mod();
    while ((int)inv.size() <= n) {
      int i = inv.size();
      inv.push_back((-inv[mod % i]) * (mod / i));
    }
    F.insert(begin(F), mint(0));
    for (int i = 1; i <= n; i++) F[i] *= inv[i];
  };

  auto inplace_diff = [](fps& F) -> void {
    if (F.empty()) return;
    F.erase(begin(F));
    mint coeff = 1, one = 1;
    for (int i = 0; i < (int)F.size(); i++) {
      F[i] *= coeff;
      coeff += one;
    }
  };

  fps b{1, 1 < (int)this->size() ? (*this)[1] : 0}, c{1}, z1, z2{1, 1};
  for (int m = 2; m < deg; m *= 2) {
    auto y = b;
    y.resize(2 * m);
    y.ntt();
    z1 = z2;
    fps z(m);
    for (int i = 0; i < m; ++i) z[i] = y[i] * z1[i];
    z.intt();
    fill(begin(z), begin(z) + m / 2, mint(0));
    z.ntt();
    for (int i = 0; i < m; ++i) z[i] *= -z1[i];
    z.intt();
    c.insert(end(c), begin(z) + m / 2, end(z));
    z2 = c;
    z2.resize(2 * m);
    z2.ntt();
    fps x(begin(*this), begin(*this) + min<int>(this->size(), m));
    inplace_diff(x);
    x.push_back(mint(0));
    x.ntt();
    for (int i = 0; i < m; ++i) x[i] *= y[i];
    x.intt();
    x -= b.diff();
    x.resize(2 * m);
    for (int i = 0; i < m - 1; ++i) x[m + i] = x[i], x[i] = mint(0);
    x.ntt();
    for (int i = 0; i < 2 * m; ++i) x[i] *= z2[i];
    x.intt();
    x.pop_back();
    inplace_integral(x);
    for (int i = m; i < min<int>(this->size(), 2 * m); ++i) x[i] += (*this)[i];
    fill(begin(x), begin(x) + m, mint(0));
    x.ntt();
    for (int i = 0; i < 2 * m; ++i) x[i] *= y[i];
    x.intt();
    b.insert(end(b), begin(x) + m, end(x));
  }
  return fps{begin(b), begin(b) + deg};
}

/**
 * @brief NTT mod用FPSライブラリ
 * @docs docs/fps/ntt-friendly-fps.md
 */

//


template <typename mint>
mint LinearRecurrence(long long k, FormalPowerSeries<mint> Q,
                      FormalPowerSeries<mint> P) {
  Q.shrink();
  mint ret = 0;
  if (P.size() >= Q.size()) {
    auto R = P / Q;
    P -= R * Q;
    P.shrink();
    if (k < (int)R.size()) ret += R[k];
  }
  if ((int)P.size() == 0) return ret;

  FormalPowerSeries<mint>::set_fft();
  if (FormalPowerSeries<mint>::ntt_ptr == nullptr) {
    P.resize((int)Q.size() - 1);
    while (k) {
      auto Q2 = Q;
      for (int i = 1; i < (int)Q2.size(); i += 2) Q2[i] = -Q2[i];
      auto S = P * Q2;
      auto T = Q * Q2;
      if (k & 1) {
        for (int i = 1; i < (int)S.size(); i += 2) P[i >> 1] = S[i];
        for (int i = 0; i < (int)T.size(); i += 2) Q[i >> 1] = T[i];
      } else {
        for (int i = 0; i < (int)S.size(); i += 2) P[i >> 1] = S[i];
        for (int i = 0; i < (int)T.size(); i += 2) Q[i >> 1] = T[i];
      }
      k >>= 1;
    }
    return ret + P[0];
  } else {
    int N = 1;
    while (N < (int)Q.size()) N <<= 1;

    P.resize(2 * N);
    Q.resize(2 * N);
    P.ntt();
    Q.ntt();
    vector<mint> S(2 * N), T(2 * N);

    vector<int> btr(N);
    for (int i = 0, logn = __builtin_ctz(N); i < (1 << logn); i++) {
      btr[i] = (btr[i >> 1] >> 1) + ((i & 1) << (logn - 1));
    }
    mint dw = mint(FormalPowerSeries<mint>::ntt_pr())
                  .inverse()
                  .pow((mint::get_mod() - 1) / (2 * N));

    while (k) {
      mint inv2 = mint(2).inverse();

      // even degree of Q(x)Q(-x)
      T.resize(N);
      for (int i = 0; i < N; i++) T[i] = Q[(i << 1) | 0] * Q[(i << 1) | 1];

      S.resize(N);
      if (k & 1) {
        // odd degree of P(x)Q(-x)
        for (auto &i : btr) {
          S[i] = (P[(i << 1) | 0] * Q[(i << 1) | 1] -
                  P[(i << 1) | 1] * Q[(i << 1) | 0]) *
                 inv2;
          inv2 *= dw;
        }
      } else {
        // even degree of P(x)Q(-x)
        for (int i = 0; i < N; i++) {
          S[i] = (P[(i << 1) | 0] * Q[(i << 1) | 1] +
                  P[(i << 1) | 1] * Q[(i << 1) | 0]) *
                 inv2;
        }
      }
      swap(P, S);
      swap(Q, T);
      k >>= 1;
      if (k < N) break;
      P.ntt_doubling();
      Q.ntt_doubling();
    }
    P.intt();
    Q.intt();
    return ret + (P * (Q.inv()))[k];
  }
}

template <typename mint>
mint kitamasa(long long N, FormalPowerSeries<mint> Q,
              FormalPowerSeries<mint> a) {
  assert(!Q.empty() && Q[0] != 0);
  if (N < (int)a.size()) return a[N];
  assert((int)a.size() >= int(Q.size()) - 1);
  auto P = a.pre((int)Q.size() - 1) * Q;
  P.resize(Q.size() - 1);
  return LinearRecurrence<mint>(N, Q, P);
}

template <class S> vector<int> kmp(const S& s) {
    int n = sz(s);
    vector<int> r(n + 1);
    r[0] = -1;
	for (int i = 0; i < n; i++) {
		int j = r[i];
		while (j >= 0 && s[i] != s[j]) {
			j = r[j];
		}
		j++;
		r[i + 1] = j;
	}
    return r;
}

int main() {
    ios_base::sync_with_stdio(false), cin.tie(nullptr);

    int M, N;
    cin >> M >> N;

    string S;
    cin >> S;
	assert(sz(S) == M);

	using mint = LazyMontgomeryModInt<998244353>;
  	using fps = FormalPowerSeries<mint>;

    auto fail = kmp(S);
	vector<bool> border(M+1);
	{
		int cur = M;
		while (true) {
			border[cur] = true;
			cur = fail[cur];
			if (cur == 0) break;
		}
	}

	constexpr int A = 26;
	fps g(M+2);
	for (int p = 0; p <= M; p++) {
		if (border[M-p]) {
			g[p] += 1;
			g[p+1] -= A;
		}
	}
	g[M] += 1;
	
	fps a(M+1);
	a[M] = 1;
	auto res = kitamasa(N, g, a);
	cout << res.get() << '\n';

    return 0;
}