#define INF 4'000'000'000'000'000'037LL
#include <bits/stdc++.h>
using namespace std;
namespace {
using ll = long long;
using uint = unsigned int;
using ull = unsigned long long;
using pll = pair<ll, ll>;
#define vc vector
template <class T>
using vvc = vc<vc<T>>;
using vpll = vc<pll>;
#ifdef __SIZEOF_INT128__
using i128 = __int128_t;
using u128 = __uint128_t;
#endif
#define cauto const auto
#define overload4(_1, _2, _3, _4, name, ...) name
#define rep1(i, n) for (ll i = 0, nnnnn = ll(n); i < nnnnn; i++)
#define rep(...) overload4(__VA_ARGS__, rep3, rep2, rep1)(__VA_ARGS__)
#define repi1(i, n) for (int i = 0, nnnnn = int(n); i < nnnnn; i++)
#define repi2(i, l, r) for (int i = int(l), rrrrr = int(r); i < rrrrr; i++)
#define repi(...) overload4(__VA_ARGS__, repi3, repi2, repi1)(__VA_ARGS__)
#define fe(...) for (auto __VA_ARGS__)
template <class T>
constexpr bool is_integral_ext = is_integral_v<T> || is_same_v<T, i128> || is_same_v<T, u128>;
template <class T>
constexpr bool is_signed_ext = is_signed_v<T> || is_same_v<T, i128>;
template <class T>
constexpr bool is_unsigned_ext = is_unsigned_v<T> || is_same_v<T, u128>;
template <class T, class U>
inline bool chmin(T &a, U b) { return a > b ? a = b, true : false; }
template <class T = ll, class U, class V, typename = enable_if_t<is_integral_ext<U> && is_integral_ext<V>>>
inline constexpr T divfloor(U a, V b) { return T(a) / T(b) - (T(a) % T(b) && (T(a) ^ T(b)) < 0); }
template <class T = ll, class U, class V, typename = enable_if_t<is_integral_ext<U> && is_integral_ext<V>>>
inline constexpr T safemod(U a, V b) { return T(a) - T(b) * divfloor<T>(a, b); }
template <class T = ll, class U, class V>
constexpr T ipow(U a, V b)
{
  assert(b >= 0);
  if (b == 0)
    return 1;
  if (a == 0 || a == 1)
    return a;
  if (a < 0 && a == -1)
    return b & 1 ? -1 : 1;
  T res = 1, tmp = a;
  while (true)
  {
    if (b & 1)
      res *= tmp;
    b >>= 1;
    if (b == 0)
      break;
    tmp *= tmp;
  }
  return res;
}
template <class T = ll, class A, class B, class M>
T mul_limited(A a, B b, M m)
{
  assert(a >= 0 && b >= 0 && m >= 0);
  if (b == 0)
    return 0;
  return T(a) > T(m) / T(b) ? T(m) : T(a) * T(b);
}
template <class T = ll, class A, class B>
T mul_limited(A a, B b) { return mul_limited<T>(a, b, INF); }
template <class T = ll, class A, class B, class M>
T pow_limited(A a, B b, M m)
{
  assert(a >= 0 && b >= 0 && m >= 0);
  if (a <= 1 || b == 0)
    return min(ipow<T>(a, b), T(m));
  T res = 1, tmp = a;
  while (true)
  {
    if (b & 1)
    {
      if (res > T(m) / tmp)
        return m;
      res *= tmp;
    }
    b >>= 1;
    if (b == 0)
      break;
    if (tmp > T(m) / tmp)
      return m;
    tmp *= tmp;
  }
  return res;
}
template <class T = ll, class A, class B>
T pow_limited(A a, B b) { return pow_limited<T>(a, b, INF); }
#define ALL(a) (a).begin(), (a).end()
template <class T = ll, class V>
inline T SZ(const V &x) { return x.size(); }
#define eb emplace_back
template <class T, size_t d, size_t i = 0, class V>
auto dvec(const V (&sz)[d], const T &init)
{
  if constexpr (i < d)
    return vc(sz[i], dvec<T, d, i + 1>(sz, init));
  else
    return init;
}
template <class T, class U>
vc<T> permuted(const vc<T> &a, const vc<U> &p)
{
  const int n = p.size();
  vc<T> res(n);
  repi(i, n)
  {
    assert(0 <= p[i] && p[i] < U(a.size()));
    res[i] = a[p[i]];
  }
  return res;
}
template <class T, class U, class... Ts>
vc<T> permuted(const vc<T> &p, const vc<U> &q, const vc<Ts> &...rs)
{
  return permuted(permuted(p, q), rs...);
}
#if __cplusplus < 202002L
#else
#endif
template <class V>
void unique(V &v) { v.erase(std::unique(ALL(v)), v.end()); }
template <class V, class U>
void rotate(V &v, U k)
{ 
  const U n = v.size();
  k = (k % n + n) % n;
  std::rotate(v.begin(), v.begin() + k, v.end());
}
template <class T>
vvc<T> top(const vvc<T> &a)
{
  if (a.empty())
    return {};
  const int n = a.size(), m = a[0].size();
  vvc<T> b(m, vc<T>(n));
  repi(i, n)
  {
    assert(SZ<int>(a[i]) == m);
    repi(j, m) b[j][i] = a[i][j];
  }
  return b;
}
template <class T, class = void>
struct has_e0 : false_type {};
template <class T>
struct has_e0<T, void_t<decltype(T::e0())>> : true_type {};
template <class T>
inline constexpr bool has_e0_v = has_e0<T>::value;
const vpll DRULgrid = {{1, 0}, {0, 1}, {-1, 0}, {0, -1}};
const vpll DRULplane = {{0, -1}, {1, 0}, {0, 1}, {-1, 0}};
template <class T>
struct is_random_access_iterator
{
  static constexpr bool value = is_same_v<
    typename iterator_traits<T>::iterator_category,
    random_access_iterator_tag
  >;
};
template <class T>
constexpr bool is_random_access_iterator_v = is_random_access_iterator<T>::value;
#if __cplusplus < 202002L
struct identity
{
  template <class T>
  constexpr T &&operator()(T &&t) const noexcept
  { return forward<T>(t); }
};
namespace internal
{
  template <class T = ll, class V, class Judge>
  inline T bound_helper(const V &v, Judge judge)
  {
    int l = -1, r = v.size();
    while (r - l > 1)
    {
      int m = (l + r) / 2;
      if (judge(m))
        l = m;
      else
        r = m;
    }
    return r;
  }
};
#else
#endif
template <class T>
inline constexpr ull MASK(T k) { return (1ULL << k) - 1ULL; }
#if __cplusplus < 202002L
inline constexpr ull countr_zero(ull x) { assert(x != 0); return __builtin_ctzll(x); }
#else
inline constexpr ll bit_width(ll x) { return std::bit_width((ull)x); }
inline constexpr ll bit_floor(ll x) { return std::bit_floor((ull)x); }
inline constexpr ll bit_ceil(ll x) { return std::bit_ceil((ull)x); }
inline constexpr ll countr_zero(ll x) { assert(x != 0); return std::countr_zero((ull)x); }
inline constexpr ll popcount(ll x) { return std::popcount((ull)x); }
inline constexpr bool has_single_bit(ll x) { return std::has_single_bit((ull)x); }
#endif
#define dump(...)
#define local(...)
#define oj(...) __VA_ARGS__
template <class T, class Sequence>
vc<T> content(queue<T, Sequence> que)
{
  vc<T> res;
  while (!que.empty())
  {
    res.eb(que.front());
    que.pop();
  }
  return res;
}
template <class T, class Sequence, class Compare>
vc<T> content(priority_queue<T, Sequence, Compare> pque)
{
  vc<T> res;
  while (!pque.empty())
  {
    res.eb(pque.top());
    pque.pop();
  }
  return res;
}
template <class T>
auto content(const T &obj) { return obj.content(); }
namespace fastio {
static constexpr uint32_t SIZ = 1 << 17;
char ibuf[SIZ];
char obuf[SIZ];
char out[100];
uint32_t pil = 0, pir = 0, por = 0;
struct Pre {
  char num[10000][4];
  constexpr Pre() : num() {
    for (int i = 0; i < 10000; i++) {
      int n = i;
      for (int j = 3; j >= 0; j--) {
        num[i][j] = n % 10 | '0';
        n /= 10;
      }
    }
  }
} constexpr pre;
inline void load() {
  memcpy(ibuf, ibuf + pil, pir - pil);
  pir = pir - pil + fread(ibuf + pir - pil, 1, SIZ - pir + pil, stdin);
  pil = 0;
  if (pir < SIZ) ibuf[pir++] = '\n';
}
inline void flush() {
  fwrite(obuf, 1, por, stdout);
  por = 0;
}
void rd1(string &x) {
  x.clear();
  char c;
  do {
    if (pil + 1 > pir) load();
    c = ibuf[pil++];
  } while (isspace(c));
  do {
    x += c;
    if (pil == pir) load();
    c = ibuf[pil++];
  } while (!isspace(c));
}
template <typename T>
void rd1_integer(T &x) {
  if (pil + 100 > pir) load();
  char c;
  do
    c = ibuf[pil++];
  while (c < '-');
  bool minus = 0;
  if constexpr (is_signed<T>::value || is_same_v<T, i128>) {
    if (c == '-') { minus = 1, c = ibuf[pil++]; }
  }
  x = 0;
  while ('0' <= c) { x = x * 10 + (c & 15), c = ibuf[pil++]; }
  pil--;
  if constexpr (is_signed<T>::value || is_same_v<T, i128>)
  {
    if (minus) x = -x;
  }
}
void rd1(ll &x) { rd1_integer(x); }
template <class T, class U>
void rd1(pair<T, U> &p) {
  return rd1(p.first), rd1(p.second);
}
template <size_t N = 0, typename T>
void rd1_tuple(T &t) {
  if constexpr (N < std::tuple_size<T>::value) {
    auto &x = std::get<N>(t);
    rd1(x);
    rd1_tuple<N + 1>(t);
  }
}
template <class... T>
void rd1(tuple<T...> &tpl) {
  rd1_tuple(tpl);
}
template <size_t N = 0, typename T>
void rd1(array<T, N> &x) {
  for (auto &d: x) rd1(d);
}
template <class T>
void rd1(vc<T> &x) {
  for (auto &d: x) rd1(d);
}
void read() {}
template <class H, class... T>
void read(H &h, T &... t) {
  rd1(h), read(t...);
}
void wt1(const char c) {
  if (por == SIZ) flush();
  obuf[por++] = c;
}
template <typename T>
void wt1_integer(T x) {
  if (por > SIZ - 100) flush();
  if (x < 0) { obuf[por++] = '-', x = -x; }
  int outi;
  for (outi = 96; x >= 10000; outi -= 4) {
    memcpy(out + outi, pre.num[x % 10000], 4);
    x /= 10000;
  }
  if (x >= 1000) {
    memcpy(obuf + por, pre.num[x], 4);
    por += 4;
  } else if (x >= 100) {
    memcpy(obuf + por, pre.num[x] + 1, 3);
    por += 3;
  } else if (x >= 10) {
    int q = (x * 103) >> 10;
    obuf[por] = q | '0';
    obuf[por + 1] = (x - q * 10) | '0';
    por += 2;
  } else
    obuf[por++] = x | '0';
  memcpy(obuf + por, out + outi + 4, 96 - outi);
  por += 96 - outi;
}
void wt1(int x) { wt1_integer(x); }
template <class T, enable_if_t<is_integral_v<T>, int> = 0>
void wt1(T x) { wt1_integer(x); }
template <class T, class U>
void wt1(const pair<T, U> &val) {
  wt1(val.first);
  wt1(' ');
  wt1(val.second);
}
template <size_t N = 0, typename T>
void wt1_tuple(const T &t) {
  if constexpr (N < std::tuple_size<T>::value) {
    if constexpr (N > 0) { wt1(' '); }
    const auto x = std::get<N>(t);
    wt1(x);
    wt1_tuple<N + 1>(t);
  }
}
template <class... T>
void wt1(const tuple<T...> &tpl) {
  wt1_tuple(tpl);
}
template <class T, size_t S>
void wt1(const array<T, S> &val) {
  auto n = val.size();
  for (size_t i = 0; i < n; i++) {
    if (i) wt1(' ');
    wt1(val[i]);
  }
}
template <class T>
void wt1(const vector<T> &val) {
  auto n = val.size();
  for (size_t i = 0; i < n; i++) {
    if (i) wt1(' ');
    wt1(val[i]);
  }
}
void print() { wt1('\n'); }
template <class Head, class... Tail>
void print(Head &&head, Tail &&... tail) {
  wt1(head);
  if (sizeof...(Tail)) wt1(' ');
  print(std::forward<Tail>(tail)...);
}
} // namespace fastio
struct Dummy {
  Dummy() { atexit(fastio::flush); }
} dummy;
namespace internal
{
template <class... Ts>
void READnodump(Ts &...a) { fastio::read(a...); }
template <class T>
void READVECnodump(int n, vc<T> &v)
{
  v.resize(n);
  READnodump(v);
}
template <class T, class... Ts>
void READVECnodump(int n, vc<T> &v, vc<Ts> &...vs)
{ READVECnodump(n, v), READVECnodump(n, vs...); }
template <class T>
void READVEC2nodump(int n, int m, vvc<T> &v)
{
  v.assign(n, vc<T>(m));
  READnodump(v);
}
template <class T, class... Ts>
void READVEC2nodump(int n, int m, vvc<T> &v, vvc<Ts> &...vs)
{ READVEC2nodump(n, m, v), READVEC2nodump(n, m, vs...); }
template <class T>
void READJAGnodump(int n, vvc<T> &v)
{
  v.resize(n);
  repi(i, n)
  {
    int k;
    READnodump(k);
    READVECnodump(k, v[i]);
  }
}
template <class T, class... Ts>
void READJAGnodump(int n, vvc<T> &v, vvc<Ts> &...vs)
{ READJAGnodump(n, v), READJAGnodump(n, vs...); }
}; // namespace internal
#define READ(...) internal::READnodump(__VA_ARGS__); dump(__VA_ARGS__)
#define IN(T, ...) T __VA_ARGS__; READ(__VA_ARGS__)
#define LL(...) IN(ll, __VA_ARGS__)
#define READVEC(...) internal::READVECnodump(__VA_ARGS__); dump(__VA_ARGS__)
#define VEC(T, n, ...) vc<T> __VA_ARGS__; READVEC(n, __VA_ARGS__)
#define PRINT fastio::print
template <class T, class U, class P>
pair<T, U> operator+=(pair<T, U> &a, const P &b)
{
  a.first += b.first;
  a.second += b.second;
  return a;
}
template <class T, class U, class P>
pair<T, U> operator+(pair<T, U> &a, const P &b) { return a += b; }
template <class T, size_t n, class A>
array<T, n> operator+=(array<T, n> &a, const A &b)
{
  for (size_t i = 0; i < n; i++)
    a[i] += b[i];
  return a;
}
template <class T, size_t n, class A>
array<T, n> operator+(array<T, n> &a, const A &b) { return a += b; }
namespace internal
{
template <size_t... I, class A, class B>
auto tuple_add_impl(A &a, const B &b, const index_sequence<I...>)
{
  ((get<I>(a) += get<I>(b)), ...);
  return a;
}
}; // namespace internal
template <class... Ts, class Tp>
tuple<Ts...> operator+=(tuple<Ts...> &a, const Tp &b)
{ return internal::tuple_add_impl(a, b, make_index_sequence<tuple_size_v<tuple<Ts...>>>{}); }
template <class... Ts, class Tp>
tuple<Ts...> operator+(tuple<Ts...> &a, const Tp &b) { return a += b; }
namespace internal
{
};
mt19937_64 mt;
template <class T>
struct larger_int;
#define LARGER_INT(T, U) \
  template <>            \
  struct larger_int<T>   \
  {                      \
    using type = U;      \
  };
LARGER_INT(signed char, short)
LARGER_INT(short, int)
LARGER_INT(int, long long)
LARGER_INT(long, __int128_t)
LARGER_INT(long long, __int128_t)
LARGER_INT(unsigned char, unsigned short)
LARGER_INT(unsigned short, unsigned int)
LARGER_INT(unsigned int, unsigned long long)
LARGER_INT(unsigned long, __uint128_t)
LARGER_INT(unsigned long long, __uint128_t)
#undef LARGER_INT
template <class T>
using larger_int_t = typename larger_int<T>::type;
namespace internal
{
template <class T>
constexpr ll powmod_constexpr(ll x, ll n, T m)
{
  if (m == 1)
    return 0;
  using U = make_unsigned_t<T>;
  using L = larger_int_t<U>;
  U r = 1, y = safemod(x, m);
  while (n)
  {
    if (n & 1)
      r = L(r) * y % m;
    y = L(y) * y % m;
    n >>= 1;
  }
  return r;
}
template <class T>
constexpr bool isprime_constexpr(T n)
{
  if constexpr (sizeof(T) > 4)
  {
    if (n <= INT_MAX)
      return isprime_constexpr<int>(n);
  }
  if (n <= 1)
    return false;
  if (n == 2 || n == 7 || n == 61)
    return true;
  if (n % 2 == 0)
    return false;
  ll d = n - 1;
  while (d % 2 == 0)
    d /= 2;
  using U = make_unsigned_t<T>;
  using L = larger_int_t<U>;
  auto miller_rabin = [&](const auto &bases) constexpr
  {
    for (ll a : bases)
    {
      ll t = d, y = powmod_constexpr(a, t, n);
      while (t != n - 1 && y != 1 && y != n - 1)
      {
        y = L(y) * y % n;
        t <<= 1;
      }
      if (y != n - 1 && t % 2 == 0)
        return false;
    }
    return true;
  };
  if constexpr(sizeof(T) <= 4)
  {
    constexpr ll bases[3] = {2, 7, 61};
    return miller_rabin(bases);
  }
  else
  {
    constexpr ll bases[7] = {2, 325, 9375, 28178, 450775, 9780504, 1795265022};
    return miller_rabin(bases);
  }
}
template <auto n>
constexpr bool isprime = isprime_constexpr(n);
};
namespace internal
{
template <auto M>
struct policy_static
{
  using mod_type = decltype(M);
  using value_type = make_unsigned_t<mod_type>;
  using calc_type = larger_int_t<value_type>;
  static constexpr bool is_prime = isprime_constexpr(M);
  static constexpr mod_type mod() { return M; }
  static constexpr value_type umod() { return M; }
  static constexpr value_type init(value_type v) { return v; }
  static constexpr mod_type val(value_type v) { return v; }
  static constexpr value_type mul(value_type a, value_type b)
  {
    return (value_type)((calc_type(a) * b) % M);
  }
};
};
namespace internal
{
struct barrett32
{
  uint m;
  ull im;
  explicit barrett32(uint m) : m(m), im((ull)(-1) / m + 1) {}
  uint umod() const { return m; }
  uint mul(uint a, uint b) const
  {
    ull z = a;
    z *= b;
    ull x = ull((u128(z) * im) >> 64);
    ull y = x * m;
    return uint(z - y + (z < y ? m : 0));
  }
};
template <int id>
struct policy_barrett32
{
  using value_type = uint;
  using calc_type = ull;
  using mod_type = int;
  static constexpr bool is_prime = false;
  static inline barrett32 reducer{998244353};
  static void set_mod(mod_type m) { reducer = barrett32(m); }
  static mod_type mod() { return reducer.umod(); }
  static value_type umod() { return reducer.umod(); }
  static value_type init(value_type v) { return v; }
  static mod_type val(value_type v) { return v; }
  static value_type mul(value_type a, value_type b) { return reducer.mul(a, b); }
};
};
namespace internal
{
inline constexpr ull inv64(ull a)
{
  ull x = a;
  while (a * x != 1) x *= 2 - a * x;
  return x;
}
struct montgomery64odd
{
  ull m, im, sq;
  explicit montgomery64odd(ull m) : m(m), im(inv64(m)), sq(-u128(m) % m) {}
  ull umod() const { return m; }
  ull reduce(u128 x) const
  {
    auto t = (x + u128(m) * (-im * ull(x))) >> 64;
    if (t >= m) t -= m;
    return (ull)t;
  }
  ull inv_reduce(i128 v) const { return reduce(u128(v % m + m) * sq); }
};
struct montgomery64
{
  ull m, mx, imx, d, q;
  uint b;
  explicit montgomery64(ull m) : m(m)
  {
    b = countr_zero(m), mx = m >> b; 
    imx = inv64(mx);
    d = powmod_constexpr((mx + 1) / 2, b, mx); 
    u128 sq = -u128(mx) % mx; 
    q = (1 + (((sq - 1) * d) << b)) % m;
  }
  ull umod() const { return m; }
  ull reduce(u128 x) const
  {
    ull p = x & MASK(b); 
    x = (x >> b) + p * d;
    ull y = p << (64 - b);
    auto t = (x + u128(mx) * (imx * (y - ull(x)))) >> (64 - b);
    if (t >= m) { t -= m; if (t >= m) t -= m; }
    return (ull)t;
  }
  ull inv_reduce(i128 v) const { return reduce(u128(v % m + m) * q); }
};
template <int id>
struct policy_montgomery64_odd
{
  using value_type = ull;
  using calc_type = u128;
  using mod_type = ll;
  static constexpr bool is_prime = false;
  static inline montgomery64odd reducer{(1LL << 61) - 1};
  static void set_mod(mod_type m) { reducer = montgomery64odd(m); }
  static mod_type mod() { return reducer.umod(); }
  static value_type umod() { return reducer.umod(); }
  static value_type init(value_type v) { return reducer.inv_reduce(v); }
  static mod_type val(value_type v) { return reducer.reduce(v); }
  static value_type mul(value_type a, value_type b) { return reducer.reduce((calc_type)a * b); }
};
template <int id>
struct policy_montgomery64
{
  using value_type = ull;
  using calc_type = u128;
  using mod_type = ll;
  static constexpr bool is_prime = false;
  static inline montgomery64 reducer{(1LL << 61) - 1};
  static void set_mod(mod_type m) { reducer = montgomery64(m); }
  static mod_type mod() { return reducer.umod(); }
  static value_type umod() { return reducer.umod(); }
  static value_type init(value_type v) { return reducer.inv_reduce(v); }
  static mod_type val(value_type v) { return reducer.reduce(v); }
  static value_type mul(value_type a, value_type b) { return reducer.reduce((calc_type)a * b); }
};
};
template <class T = ll>
constexpr tuple<T, T, T> extgcd(T a, T b)
{
  if (a == 0 && b == 0)
    return {0, 0, 0};
  T x1 = 1, y1 = 0, z1 = a;
  T x2 = 0, y2 = 1, z2 = b;
  while (z2 != 0)
  {
    T q = z1 / z2;
    tie(x1, x2) = make_pair(x2, x1 - q * x2);
    tie(y1, y2) = make_pair(y2, y1 - q * y2);
    tie(z1, z2) = make_pair(z2, z1 - q * z2);
  }
  if (z1 < 0)
    x1 = -x1, y1 = -y1, z1 = -z1;
  return {z1, x1, y1};
}
namespace internal
{
  template <class Policy>
  struct modint_impl
  {
    using V = typename Policy::value_type;
    using M = typename Policy::mod_type;
    using mint = modint_impl;
  private:
    V _v;
  public:
    static constexpr M mod() { return Policy::mod(); }
    template <class T = Policy>
    static auto set_mod(M m) -> decltype(T::set_mod(m)) { return T::set_mod(m); }
    static mint raw(V v)
    {
      mint x;
      x._v = v;
      return x;
    }
    modint_impl() : _v(0) {}
    template <class T, typename = enable_if_t<is_integral_ext<T>>>
    modint_impl(T v)
    {
      V rem;
      if constexpr (is_signed_ext<T>)
      {
        using S = make_signed_t<V>;
        S x = v % S(Policy::umod());
        if (x < 0)
          x += Policy::umod();
        rem = x;
      }
      else
        rem = V(v % Policy::umod());
      _v = Policy::init(rem);
    };
    M val() const { return Policy::val(_v); }
    mint &operator+=(const mint &rhs)
    {
      _v += rhs._v;
      if (_v >= Policy::umod())
        _v -= Policy::umod();
      return *this;
    }
    mint &operator-=(const mint &rhs)
    {
      _v -= rhs._v;
      if (_v >= Policy::umod())
        _v += Policy::umod();
      return *this;
    }
    mint &operator*=(const mint &rhs)
    {
      _v = Policy::mul(_v, rhs._v);
      return *this;
    }
    mint &operator/=(const mint &rhs)
    {
      return *this *= rhs.inv();
    }
    mint &operator++()
    {
      _v++;
      if (_v == Policy::umod())
        _v = 0;
      return *this;
    }
    mint &operator--()
    {
      if (_v == 0)
        _v = Policy::umod();
      _v--;
      return *this;
    }
    mint operator++(int)
    {
      mint res = *this;
      ++(*this);
      return res;
    }
    mint operator--(int)
    {
      mint res = *this;
      --(*this);
      return res;
    }
    mint operator+() const { return *this; }
    mint operator-() const { return mint() - *this; }
    template <class T>
    mint pow(T n) const
    {
      assert(n >= 0);
      mint x = *this, r = 1;
      while (n)
      {
        if (n & 1)
          r *= x;
        x *= x;
        n >>= 1;
      }
      return r;
    }
    mint inv() const
    {
      if constexpr (Policy::is_prime)
      {
        return pow(mod() - 2);
      }
      else
      {
        auto [g, x, y] = extgcd<M>(val(), mod());
        assert(g == 1);
        return mint(x);
      }
    }
    friend mint operator+(const mint &lhs, const mint &rhs) { return mint(lhs) += rhs; }
    friend mint operator-(const mint &lhs, const mint &rhs) { return mint(lhs) -= rhs; }
    friend mint operator*(const mint &lhs, const mint &rhs) { return mint(lhs) *= rhs; }
    friend mint operator/(const mint &lhs, const mint &rhs) { return mint(lhs) /= rhs; }
    friend bool operator==(const mint &lhs, const mint &rhs) { return lhs._v == rhs._v; }
    friend bool operator!=(const mint &lhs, const mint &rhs) { return lhs._v != rhs._v; }
    friend void rd1(mint &x)
    {
      long long a;
      fastio::rd1(a);
      x = a;
    }
    friend void wt1(const mint &x)
    {
      fastio::wt1(x.val());
    }
  };
};
template <int mod>
using static_modint32 = internal::modint_impl<internal::policy_static<mod>>;
template <int id>
using dynamic_modint32 = internal::modint_impl<internal::policy_barrett32<id>>;
template <ll mod>
using static_modint64 = internal::modint_impl<internal::policy_static<mod>>;
template <int id>
using dynamic_modint64_odd = internal::modint_impl<internal::policy_montgomery64_odd<id>>;
template <int id>
using dynamic_modint64 = internal::modint_impl<internal::policy_montgomery64<id>>;
using modint1000000007 = static_modint32<1000000007>;
template <class T>
struct is_modint : std::false_type
{
};
template <class Policy>
struct is_modint<internal::modint_impl<Policy>> : std::true_type
{
};
template <class T>
inline constexpr bool is_modint_v = is_modint<T>::value;
template <class T>
struct is_static_modint : false_type {};
template <int m>
struct is_static_modint<static_modint32<m>> : true_type {};
template <ll m>
struct is_static_modint<static_modint64<m>> : true_type {};
template <class T>
inline constexpr bool is_static_modint_v = is_static_modint<T>::value;
template <class T>
struct is_dynamic_modint : false_type {};
template <int id>
struct is_dynamic_modint<dynamic_modint32<id>> : true_type {};
template <int id>
struct is_dynamic_modint<dynamic_modint64_odd<id>> : true_type {};
template <int id>
struct is_dynamic_modint<dynamic_modint64<id>> : true_type {};
template <class T>
inline constexpr bool is_dynamic_modint_v = is_dynamic_modint<T>::value;
template <typename, typename = void>
struct has_mod : std::false_type
{
};
template <typename T>
struct has_mod<T, std::void_t<decltype(T::mod())>> : std::true_type
{
};
template <class mint>
struct PowerTable
{
private:
  decltype(mint::mod()) mod;
  mint base;
  vc<mint> pw;
public:
  PowerTable() {}
  PowerTable(const mint &base) : mod(mint::mod()), base(base), pw(1, 1) {}
  void reserve(int n)
  {
    if (mod != mint::mod())
    {
      mod = mint::mod();
      pw = {1};
    }
    int i = pw.size();
    if (n < i)
      return;
    pw.resize(n + 1);
    for (; i <= n; i++)
      pw[i] = pw[i - 1] * base;
  }
  mint pow(int n)
  {
    reserve(n);
    return pw[n];
  }
};
template <class T>
struct Binomial
{
private:
  inline static decltype(T::mod()) mod;
public:
  inline static vc<T> fac_, finv_, inv_;
  static void reserve(int n)
  {
    if constexpr (is_dynamic_modint_v<T>)
    {
      if (mod != T::mod())
      {
        mod = T::mod();
        fac_ = {1, 1}, finv_ = {1, 1}, inv_ = {0, 1};
      }
    }
    else
    {
      if (fac_.empty())
        fac_ = {1, 1}, finv_ = {1, 1}, inv_ = {0, 1};
    }
    if (n < SZ(fac_))
      return;
    chmin(n, T::mod() - 1);
    int si = fac_.size();
    fac_.resize(n + 1), finv_.resize(n + 1), inv_.resize(n + 1);
    repi(i, si, n + 1)
    {
      fac_[i] = fac_[i - 1] * T::raw(i);
      inv_[i] = -inv_[T::mod() % i] * T::raw(T::mod() / i);
      finv_[i] = finv_[i - 1] * inv_[i];
    }
  }
  static T inv(int n)
  {
    assert(n != 0);
    reserve(n);
    return inv_[n];
  }
};
#if __cplusplus >= 202302L
namespace cpp_dump
{
  struct mint_to_rat_fn
  {
    template <class T>
    constexpr auto operator()(const T &x) const -> decltype(mint_to_rat(x))
    {
      return mint_to_rat(x);
    }
  };
  struct rat_closure : std::ranges::range_adaptor_closure<rat_closure>
  {
    template <typename T>
    constexpr auto operator()(T &&t) const
    {
      if constexpr (!std::ranges::range<T> && std::invocable<mint_to_rat_fn, decltype(std::forward<T>(t))>)
      {
        return mint_to_rat_fn{}(std::forward<T>(t));
      }
      else if constexpr (std::ranges::range<T>)
      {
        using Ref = std::ranges::range_reference_t<T>;
        if constexpr (std::invocable<mint_to_rat_fn, decltype(std::forward<Ref>(std::declval<Ref>()))>)
        {
          return std::forward<T>(t) | std::views::transform(mint_to_rat_fn{});
        }
        else if constexpr (std::ranges::range<Ref>)
        {
          return std::forward<T>(t) | std::views::transform([this](auto &&inner)
                                                            { return (*this)(std::forward<decltype(inner)>(inner)); });
        }
        else
        {
          static_assert(false);
        }
      }
      else
      {
        static_assert(false);
      }
    }
  };
  template <typename T>
    requires(!std::ranges::range<T> && std::invocable<mint_to_rat_fn, T>)
  constexpr auto operator|(T &&t, const rat_closure &c)
  {
    return c(std::forward<T>(t));
  }
}
#endif
using mint = modint1000000007;
void init()
{
  oj(mt.seed(random_device()()));
}
void main2()
{
  LL(H, W);
  VEC(string, H, S);
  auto dp = dvec({H, H}, mint(0));
  auto ndp = dvec({H, H}, mint(0));
  dp[0][H - 1] = (S[0][0] == S[H - 1][W - 1]);
  rep(k, (H + W - 2) / 2)
  {
    rep(i1, H) rep(i2, H) ndp[i1][i2] = 0;
    rep(i1, H) rep(i2, H)
    {
      if (dp[i1][i2] == 0)
        continue;
      ll j1 = k - i1;
      ll j2 = H + W - 2 - k - i2;
      dump(i1, j1, i2, j2);
      fe([di1, dj1] : vpll{{0, 1}, {1, 0}})
      {
        fe([di2, dj2] : vpll{{0, -1}, {-1, 0}})
        {
          ll ni1 = i1 + di1, ni2 = i2 + di2;
          ll nj1 = j1 + dj1, nj2 = j2 + dj2;
          if (!(0 <= ni1 && ni1 < H))
            continue;
          if (!(0 <= ni2 && ni2 < H))
            continue;
          if (!(0 <= nj1 && nj1 < W))
            continue;
          if (!(0 <= nj2 && nj2 < W))
            continue;
          if (S[ni1][nj1] != S[ni2][nj2])
            continue;
          dump(ni1, nj1, ni2, nj2);
          ndp[ni1][ni2] += dp[i1][i2];
        }
      }
    }
    swap(dp, ndp);
    local(dump(k); rep(i1, H) rep(i2, H) if (dp[i1][i2].val()) dump(i1, i2, dp[i1][i2]));
  }
  if ((H + W - 1) % 2 == 0)
  {
    mint ans = 0;
    rep(i1, H) rep(i2, H)
    {
      if (dp[i1][i2] == 0)
        continue;
      const ll k = (H + W - 2) / 2;
      ll j1 = k - i1;
      ll j2 = H + W - 2 - k - i2;
      if (abs(i1 - i2) + abs(j1 - j2) == 1 && S[i1][j1] == S[i2][j2])
      {
        dump(i1, j1, i2, j2, dp[i1][i2]);
        ans += dp[i1][i2];
      }
    }
    PRINT(ans);
  }
  else
  {
    mint ans = 0;
    rep(i, H) ans += dp[i][i];
    PRINT(ans);
  }
}
void test()
{
}
template <auto init, auto main2, auto test>
struct Main
{
  Main()
  {
    cauto CERR = [](string val, string color)
    {
      string s = "\033[" + color + "m" + val + "\033[m";
      /* コードテストで確認する際にコメントアウトを外す
      cerr << val;
      //*/
    };
    CERR("\n[FAST_IO]\n\n", "32");
    cout << fixed << setprecision(20);
    init();
    CERR("\n[SINGLE_TESTCASE]\n\n", "36");
    main2();
  }
};
Main<init, main2, test> main_dummy;
}
int main() {}