#line 1 "No_179_\u5857\u308a\u5206\u3051.cpp"
#define YRSD
// #include "YRS/aa/fast.hpp"
#line 2 "YRS/all.hpp"

#line 2 "YRS/aa/head.hpp"

#include <iostream>
#include <algorithm>

#include <array>
#include <bitset>
#include <map>
#include <numeric>
#include <queue>
#include <set>
#include <string>
#include <tuple>

#include <bit>
#include <chrono>
#include <functional>
#include <iomanip>
#include <utility>
#include <type_traits>
#include <cassert>
#include <cctype>
#include <cmath>
#include <cstring>
#include <ctime>
#include <limits>
#include <ranges>
#include <concepts>

#define TE template <typename T>
#define TES template <typename T, typename ...S>
#define Z auto
#define ep emplace_back
#define eb emplace
#define fi first
#define se second
#define all(x) (x).begin(), (x).end()

#define OV4(a, b, c, d, e, ...) e
#define FOR1(a) for (int _ = 0; _ < (a); ++_)
#define FOR2(i, a) for (int i = 0; i < (a); ++i)
#define FOR3(i, a, b) for (int i = (a); i < (b); ++i)
#define FOR4(i, a, b, c) for (int i = (a); i < (b); i += (c))
#define FOR(...) OV4(__VA_ARGS__, FOR4, FOR3, FOR2, FOR1)(__VA_ARGS__)
#define FOR1_R(a) for (int _ = (a) - 1; _ >= 0; --_)
#define FOR2_R(i, a) for (int i = (a) - 1; i >= 0; --i)
#define FOR3_R(i, a, b) for (int i = (b) - 1; i >= (a); --i)
#define FOR4_R(i, a, b, c) for (int i = (b) - 1; i >= (a); i -= (c))
#define FOR_R(...) OV4(__VA_ARGS__, FOR4_R, FOR3_R, FOR2_R, FOR1_R)(__VA_ARGS__)
#define FOR_subset(t, s) for (int t = (s); t > -1; t = (t == 0 ? -1 : (t - 1) & s))

#define sort ranges::sort

using namespace std;

TE using vc = vector<T>;
TE using vvc = vc<vc<T>>;
TE using T1 = tuple<T>;
TE using T2 = tuple<T, T>;
TE using T3 = tuple<T, T, T>;
TE using T4 = tuple<T, T, T, T>;
TE using max_heap = priority_queue<T>;
TE using min_heap = priority_queue<T, vc<T>, greater<T>>;
using u8 = unsigned char; using uint = unsigned int; using ll = long long;      using ull = unsigned long long;
using ld = long double;   using i128 = __int128;     using u128 = __uint128_t;  using f128 = __float128;
using u16 = uint16_t;
using PII = pair<int, int>;   using PLL = pair<ll, ll>;

#ifdef YRSD
constexpr bool dbg = 1;
#else
constexpr bool dbg = 0;
#endif
#line 2 "YRS/IO/IO.hpp"

istream &operator>>(istream &I, i128 &x) {
  static string s;
  I >> s;
  int f = s[0] == '-';
  x = 0;
  const int N = (int)s.size();
  FOR(i, f, N) x = x * 10 + s[i] - '0';
  if (f) x = -x;
  return I;
}
ostream &operator<<(ostream &O, i128 x) {
  static string s;
  s.clear();
  bool f = x < 0;
  if (f) x = -x;
  while (x) s += '0' + x % 10, x /= 10;
  if (s.empty()) s += '0';
  if (f) s += '-';
  reverse(all(s));
  return O << s;
}
istream &operator>>(istream &I, f128 &x) {
  static string s;
  I >> s, x = stold(s);
  return I;
}
ostream &operator<<(ostream &O, const f128 x) { return O << ld(x); }
template <typename... S>
istream &operator>>(istream &I, tuple<S...> &t) {
  return apply([&I](Z &...s) { ((I >> s), ...); }, t), I;
}
template <typename T, typename U>
istream &operator>>(istream &I, pair<T, U> &x) {
  return I >> x.fi >> x.se;
}
template <typename T, typename U>
ostream &operator<<(ostream &O, const pair<T, U> &x) {
  return O << x.fi << ' ' << x.se;
}
TE requires requires(T &c) { begin(c); end(c); } and 
                          (not is_same_v<decay_t<T>, string>)
istream &operator>>(istream &I, T &c) {
  for (Z &e : c) I >> e;
  return I;
}
TE requires requires(const T &c) { begin(c); end(c); } and 
  (not is_same_v<decay_t<T>, const char*>) and 
  (not is_same_v<decay_t<T>, string>) and 
  (not is_array_v<remove_reference_t<T>> or 
   not is_same_v<remove_extent_t<remove_reference_t<T>>, char>)
ostream &operator<<(ostream &O, const T &a) {
  if (a.empty()) return O;
  Z i = a.begin();
  O << *i++;
  for (; i != a.end(); ++i) O << ' ' << *i;
  return O;
}
void IN() {}
TE void IN(T &x, Z &...s) { cin >> x, IN(s...); }
void print() { cout << '\n'; }
TES void print(T &&x, S &&...y) {
  cout << x;
  if constexpr (sizeof...(S)) cout << ' ';
  print(forward<S>(y)...);
}
void put() {}
TES void put(T &&x, S &&...y) {
  cout << x;
  put(forward<S>(y)...);
}

#define INT(...)  int    __VA_ARGS__; IN(__VA_ARGS__)
#define UINT(...) uint   __VA_ARGS__; IN(__VA_ARGS__)
#define LL(...)   ll     __VA_ARGS__; IN(__VA_ARGS__)
#define ULL(...)  ull    __VA_ARGS__; IN(__VA_ARGS__)
#define I128(...) i128   __VA_ARGS__; IN(__VA_ARGS__)
#define STR(...)  string __VA_ARGS__; IN(__VA_ARGS__)
#define CH(...)   char   __VA_ARGS__; IN(__VA_ARGS__)
#define REAL(...) re     __VA_ARGS__; IN(__VA_ARGS__)
#define VEC(T, a, n) vc<T> a(n); IN(a)

void YES(bool o = 1) { print(o ? "YES" : "NO"); }
void Yes(bool o = 1) { print(o ? "Yes" : "No"); }
void yes(bool o = 1) { print(o ? "yes" : "no"); }
void NO(bool o = 1) { YES(not o); }
void No(bool o = 1) { Yes(not o); }
void no(bool o = 1) { yes(not o); }
void ALICE(bool o = 1) { print(o ? "ALICE" : "BOB"); }
void Alice(bool o = 1) { print(o ? "Alice" : "Bob"); }
void alice(bool o = 1) { print(o ? "alice" : "bob"); }
void BOB(bool o = 1) { ALICE(not o); }
void Bob(bool o = 1) { Alice(not o); }
void bob(bool o = 1) { alice(not o); }
void POSSIBLE(bool o = 1) { print(o ? "POSSIBLE" : "IMPOSSIBLE"); }
void Possible(bool o = 1) { print(o ? "Possible" : "Impossible"); }
void possible(bool o = 1) { print(o ? "possible" : "impossible"); }
void IMPOSSIBLE(bool o = 1) { POSSIBLE(not o); }
void Impossible(bool o = 1) { Possible(not o); }
void impossible(bool o = 1) { possible(not o); }
void TAK(bool o = 1) { print(o ? "TAK" : "NIE"); }
void NIE(bool o = 1) { TAK(not o); }
#line 5 "YRS/all.hpp"

#if (__cplusplus >= 202002L)
#include <numbers>
constexpr ld pi = numbers::pi_v<ld>;
#endif
TE constexpr T inf = numeric_limits<T>::max();
template <> constexpr i128 inf<i128> = i128(inf<ll>) * 2'000'000'000'000'000'000;
template <typename T, typename U>
constexpr pair<T, U> inf<pair<T, U>> = {inf<T>, inf<U>};

TE constexpr static inline int pc(T x) { return popcount(make_unsigned_t<T>(x)); }
constexpr static inline ll len(const Z &a) { return a.size(); }

void reverse(Z &a) { reverse(all(a)); }

void unique(Z &a) {
  sort(a);
  a.erase(unique(all(a)), a.end());
}
TE vc<int> inverse(const vc<T> &a) {
  int N = len(a);
  vc<int> b(N, -1);
  FOR(i, N) if (a[i] != -1) b[a[i]] = i;
  return b;
}

Z QMAX(const Z &a) { return *max_element(all(a)); }
Z QMIN(const Z &a) { return *min_element(all(a)); }
TE Z QMAX(T l, T r) { return *max_element(l, r); }
TE Z QMIN(T l, T r) { return *min_element(l, r); }
constexpr bool chmax(Z &a, const Z &b) { return (a < b ? a = b, 1 : 0); }
constexpr bool chmin(Z &a, const Z &b) { return (a > b ? a = b, 1 : 0); }

vc<int> argsort(const Z &a) {
  vc<int> I(len(a));
  iota(all(I), 0);
  sort(I, [&](int i, int k) { return a[i] < a[k] or (a[i] == a[k] and i < k); });
  return I;
}
TE vc<T> rearrange(const vc<T> &a, const vc<int> &I) {
  int N = len(I);
  vc<T> b(N);
  FOR(i, N) b[i] = a[I[i]];
  return b;
}
template <int of = 1, typename T> 
vc<T> pre_sum(const vc<T> &a) {
  int N = len(a);
  vc<T> c(N + 1);
  FOR(i, N) c[i + 1] = c[i] + a[i];
  if (of == 0) c.erase(c.begin());
  return c;
}

TE constexpr static int topbit(T x) {
  if (x == 0) return - 1;
  if constexpr (sizeof(T) <= 4) return 31 - __builtin_clz(x);
  else return 63 - __builtin_clzll(x);
}
TE constexpr static int lowbit(T x) {
  if (x == 0) return -1;
  if constexpr (sizeof(T) <= 4) return __builtin_ctz(x);
  else return __builtin_ctzll(x);
}

TE constexpr T floor(T x, T y) { return x / y - (x % y and (x ^ y) < 0); }
TE constexpr T ceil(T x, T y) { return floor(x + y - 1, y); }
TE constexpr T bmod(T x, T y) { return x - floor(x, y) * y; }
TE constexpr pair<T, T> divmod(T x, T y) {
  T q = floor(x, y);
  return pair{q, x - q * y};
}
template <typename T = ll>
T SUM(const Z &v) {
  return accumulate(all(v), T(0));
}
int lb(const Z &a, Z x) { return lower_bound(all(a), x) - a.begin(); }
TE int lb(T l, T r, Z x) { return lower_bound(l, r, x) - l; }
int ub(const Z &a, Z x) { return upper_bound(all(a), x) - a.begin(); }
TE int ub(T l, T r, Z x) { return upper_bound(l, r, x) - l; }

template <bool ck = 1>
ll bina(Z f, ll l, ll r) {
  if constexpr (ck) assert(f(l));
  while (abs(l - r) > 1) {
    ll x = (r + l) >> 1;
    (f(x) ? l : r) = x;
  }
  return l;
}
TE T bina_real(Z f, T l, T r, int c = 100) {
  while (c--) {
    T x = (l + r) / 2;
    (f(x) ? l : r) = x;
  }
  return (l + r) / 2;
}

Z pop(Z &s) {
  if constexpr (requires { s.pop_back(); }) {
    Z x = s.back();
    return s.pop_back(), x;
  } else if constexpr (requires { s.top(); }) {
    Z x = s.top();
    return s.pop(), x;
  } else {
    Z x = s.front();
    return s.pop(), x;
  }
}
void setp(int x) { cout << fixed << setprecision(x); }

TE inline void sh(vc<T> &a, int N, T b = {}) {
  a.resize(N, b);
}
#line 1 "YRS/debug.hpp"
#ifdef YRSD
void DBG() { cerr << "]" << endl; }
TES void DBG(T &&x, S &&...y) {
  cerr << x;
  if constexpr (sizeof...(S)) cerr << ", ";
  DBG(forward<S>(y)...);
}
#define debug(...) cerr << "[" << __LINE__ << "]: [" #__VA_ARGS__ "] = [", DBG(__VA_ARGS__)
void ERR() { cerr << endl; }
TES void ERR(T &&x, S &&...y) {
  cerr << x;
  if constexpr (sizeof...(S)) cerr << ", ";
  ERR(forward<S>(y)...);
}
#define err(...) cerr << "[" << __LINE__ << "]: ", ERR(__VA_ARGS__)
#define asser assert
#else
#define debug(...) void(0721)
#define err(...)   void(0721)
#endif
#line 5 "No_179_\u5857\u308a\u5206\u3051.cpp"
// #include "YRS/IO/fast_io.hpp"
// #include "YRS/random/rng.hpp"
#line 2 "YRS/ds/basic/retsu.hpp"

TE struct retsu {
  int N, M;
  vc<T> a;

  retsu(int N, int M, T bs = T()) : N(N), M(M), a(N * M, bs) {}

  T* operator[](int i) { return a.data() + i * M; }
  const T* operator[](int i) const { return a.data() + i * M; }

  void fill(T x) { std::fill(all(a), x); }

  T max() const { return QMAX(a); }

  T min() const { return QMIN(a); }

  inline int id(int x, int y) const { return x * M + y; }

  void pres() {
    FOR(i, 1, N) FOR(k, M) a[id(i, k)] += a[id(i - 1, k)];
    FOR(i, N) FOR(k, 1, M) a[id(i, k)] += a[id(i, k - 1)];
  }

  inline T prod(int x, int y) const { return a[id(x, y)]; }

  inline T prod(int l, int r, int a, int b) const {
    return prod(r, b) + prod(l, a) - prod(l, b) - prod(r, a);
  }

  vc<vc<T>> to_v() const {
    vector c(N, vc<T>(M));
    FOR(i, N) FOR(k, M) c[i][k] = a[id(i, k)];
    return c;
  }
};

TE istream &operator>>(istream &I, retsu<T> &a) {
  for (Z &e : a.a) I >> e;
  return I;
}
TE ostream &operator<<(ostream &O, retsu<T> &a) {
  FOR(i, a.N) FOR(k, a.M) O << a[i][k] << " \n"[k + 1 == a.M and i + 1 != a.N];
  return O;
}

#ifdef FIO
TE inline void rd(retsu<T> &a) {
  for (T &x : a.a) rd(x);
}
TE inline void wt(retsu<T> &a) {
  FOR(i, a.N) {
    FOR(k, a.M) {
      if (k) wt(' ');
      wt(a[i][k]);
    }
    if (i != a.M) wt('\n');
  }
}
#endif
#line 8 "No_179_\u5857\u308a\u5206\u3051.cpp"
// #include "YRS/mod/mint.hpp"
// #include "YRS/aa/def.hpp"
#line 2 "YRS/flow/BM.hpp"

#line 2 "YRS/g/01coloring.hpp"

#line 2 "YRS/ds/un/dsu.hpp"

struct dsu {
  int c;
  vc<int> fa;
  dsu(int N = 0) : c(N), fa(N, -1) {}

  int f(int x) {
    while (fa[x] >= 0) {
      int p = fa[fa[x]];
      if (p < 0) return fa[x];
      x = fa[x] = p;
    }
    return x;
  }

  int operator[](int x) { return f(x); }

  bool merge(int x, int y) {
    x = f(x), y = f(y);
    if (x == y) return false;
    if (fa[x] > fa[y]) swap(x, y);
    fa[x] += fa[y];
    fa[y] = x;
    --c;
    return true;
  }

  // set fa[y] = x;
  bool set(int x, int y) {
    x = f(x), y = f(y);
    if (x == y) return false;
    fa[x] += fa[y];
    fa[y] = x;
    --c;
    return true;
  }

  int size(int x) { return -fa[f(x)]; }

  int count() const { return c; }

  bool same(int x, int y) { return f(x) == f(y); }

  void build(int N) { fa.assign(N, -1), c = N; }

  void reset() { std::fill(all(fa), -1), c = len(fa); }

  vc<vc<int>> get_group() {
    int N = len(fa);
    vc<vc<int>> v(N), s;
    FOR(i, N) v[f(i)].ep(i);
    FOR(i, N) if (not v[i].empty()) s.ep(v[i]);
    return s;
  }

  void pr() {
    const int N = len(fa);
    vc<int> res(N);
    FOR(i, N) res[i] = f(i);
    print("fa:", res);
  }
};
#line 2 "YRS/g/Basic.hpp"

#line 2 "YRS/ds/basic/hashmap.hpp"

#line 2 "YRS/random/hash.hpp"

#line 2 "YRS/random/rng.hpp"

#include <random>

#ifdef MeIoN
std::mt19937 rg(0);
std::mt19937_64 rd_64(0);
#else
std::mt19937 rg(std::chrono::steady_clock::now().time_since_epoch().count());
std::mt19937_64 rd_64(std::chrono::steady_clock::now().time_since_epoch().count());
#endif

uint rng() { return rg(); }
uint rng(uint lim) { return rg() % lim; }
int rng(int l, int r) { return l + rg() % (r - l); }
ull rng_64() { return rd_64(); }
ull rng_64(ull lim) { return rd_64() % lim; }
ll rng_64(ll l, ll r) { return l + rd_64() % (r - l); }

template <typename T>
void shuffle(vector<T> &v) {
  const int N = len(v);
  FOR(i, 1, N) {
    int k = rng(0, i + 1);
    if (i != k) swap(v[i], v[k]);
  }
}
#line 4 "YRS/random/hash.hpp"

TE ull hsh(const pair<T, T> &X) {
  static ull B = rng_64();
  if (not B) B = rng_64();
  return B * X.fi + X.se;
}
#line 4 "YRS/ds/basic/hashmap.hpp"

// https://judge.yosupo.jp/problem/associative_array

// reserve N 时最快
TE struct hashmap {
  uint ls, msk;
  vc<ull> ke;
  vc<T> val;
  vc<u8> vis;

  ull hash(ull x) const {
    static const ull bs =
        chrono::steady_clock::now().time_since_epoch().count();
    x += bs;
    x = (x ^ (x >> 30)) * 0xbf58476d1ce4e5b9;
    x = (x ^ (x >> 27)) * 0x94d049bb133111eb;
    return (x ^ (x >> 31)) & msk;
  }

  void extend() {
    vc<pair<ull, T>> dat;
    const int N = len(vis);
    dat.reserve(N / 2 - ls);
    FOR(i, N) if (vis[i]) dat.ep(ke[i], val[i]);
    build(dat.size() << 1);
    for (Z &[a, b] : dat) (*this)[a] = b;
  }

  hashmap(uint N = 0) { build(N); }

  void build(uint N) {
    uint k = 8;
    while (k < (N << 1)) k <<= 1;
    ls = k >> 1, msk = k - 1;
    ke.resize(k);
    val.resize(k);
    vis.assign(k, 0);
  }

  void clear() {
    fill(all(vis), 0);
    ls = (msk + 1) >> 1;
  }

  ll size() const { return vis.size() / 2 - ls; }

  int id(ull k) const {
    int i = hash(k);
    while (vis[i] and ke[i] != k) i = (i + 1) & msk;
    return i;
  }

  T &operator[](ull k) {
    if (ls == 0) extend();
    int i = id(k);
    if (not vis[i]) {
      vis[i] = 1;
      ke[i] = k;
      val[i] = T {};
      --ls;
    }
    return val[i];
  }
  
  T &operator[](PII p) {
    ll k = hsh(p);
    if (ls == 0) extend();
    int i = id(k);
    if (not vis[i]) {
      vis[i] = 1;
      ke[i] = k;
      val[i] = T {};
      --ls;
    }
    return val[i];
  }

  T get(ull k, T fail) const {
    int i = id(k);
    return (vis[i] ? val[i] : fail);
  }

  bool contains(ull k) const {
    int i = id(k);
    return vis[i] and ke[i] == k;
  }

  vc<pair<ull, T>> get_all() const {
    int N = len(vis);
    vc<pair<ull, T>> s;
    FOR(i, N) if (vis[i]) s.ep(ke[i], val[i]);
    return s;
  }

  // f(ke, val);
  void enumerate_all(Z f) const {
    const int N = len(vis);
    FOR(i, N) if (vis[i]) f(ke[i], val[i]);
  }
};
#line 4 "YRS/g/Basic.hpp"

// https://www.luogu.com.cn/problem/P5318

TE struct edge {
  int f, to;
  T w;
  int id;
};
template <typename T = int, bool dir = 0>
struct graph {
  static constexpr bool is_dir = dir;
  int N, M;
  using cost_type = T;
  using ee = edge<T>;
  vc<ee> es;
  vc<int> in;
  vc<ee> c;
  
  bool ok;

  bool isok() { return ok; }

  struct px {
    const graph *g;
    int l, r;
    px(const graph *g, int l, int r) : g(g), l(l), r(r) {}
    const ee *begin() const {
      if (l == r) return 0;
      return &g->c[l];
    }
    const ee *end() const {
      if (l == r) return 0;
      return &g->c[r];
    }
  };

  px operator[](int i) const {
    assert(ok);
    return {this, in[i], in[i + 1]};
  }

  graph() : N(0), M(0), ok(0) {}
  graph(int N) : N(N), M(0), ok(0) {}

  void add(int f, int t, T w = 1, int i = -1) {
    assert(not ok);
    assert(-1 < f and -1 < t and t < N and f < N);
    if (i == -1) i = M;
    Z e = ee({f, t, w, i});
    es.ep(e);
    ++M;
  }

  void build() {
    assert(not ok);
    ok = 1;
    in.assign(N + 1, 0);
    for (Z &&e : es) {
      in[e.f + 1]++;
      if (not dir) in[e.to + 1]++;
    }
    FOR(i, N) in[i + 1] += in[i];
    Z cc = in;
    c.resize(in.back() + 1);
    for (Z &&e : es) {
      c[cc[e.f]++] = e;
      if (not dir) c[cc[e.to]++] = ee({e.to, e.f, e.w, e.id});
    }
  }

  template <bool wt = 0, int of = 1>
  void sc() { sc<wt, of>(N - 1); }
  template <bool wt = 0, int of = 1>
  void sc(int M) {
    es.reserve(M * (dir ? 1 : 2));
    FOR(M) {
      INT(x, y);
      x -= of, y -= of;
      if (not wt) {
        add(x, y);
      } else {
        T w;
        IN(w);
        add(x, y, w);
      }
    }
    build();
  }

  vc<int> deg() {
    vc<int> in(N);
    for (Z &&e : es) ++in[e.f], ++in[e.to];
    return in;
  }

  pair<vc<int>, vc<int>> deg_inout() {
    vc<int> in(N), ou(N);
    for (Z &&e : es) ++in[e.to], ++ou[e.f];
    return {in, ou};
  }

  vc<int> ni;
  vc<u8> vis;
  // 使G中的顶点V[i]在新图表中为i
  // {G, es}
  // sum（deg(v)）的计算量
  // 注意它可能大于新图表的n+M
  graph<T, dir> rearrange(const vc<int> &v, bool keep_eid = 0) {
    if (len(ni) != N) ni.assign(N, -1);
    int N = len(v);
    FOR(i, N) ni[v[i]] = i;
    graph<T, dir> g(N);
    vc<int> s;
    FOR(i, N) {
      for (Z &&e : (*this)[v[i]]) {
        if (len(vis) <= e.id) vis.resize(e.id + 1);
        if (vis[e.id]) continue;
        int f = e.f, to = e.to;
        if (ni[f] != -1 and ni[to] != -1) {
          s.ep(e.id);
          vis[e.id] = 1;
          int id = (keep_eid ? e.id : -1);
          g.add(ni[f], ni[to], e.w, id);
        }
      }
    }
    FOR(i, N) ni[v[i]] = -1;
    for (int i : s) vis[i] = 0;
    return g.build(), g;
  }

  ull has(ull x, ull y) {
    if (not dir and x > y) swap(x, y);
    return x * N + y;
  }

  hashmap<int> mp;
  int get_eid(ull x, ull y) {
    if (mp.size() == 0) {
      mp.build(N - 1);
      for (Z &&e : es) {
        ull x = e.f, y = e.to;
        ull k = has(x, y);
        mp[k] = e.id;
      }
    }
    return mp.get(has(x, y), -1);
  }

  graph rev() const {
    static_assert(dir);
    graph ng(N);
    for (Z &&[f, t, w, id] : es) ng.add(t, f, w, id);
    return ng;
  }
};
#line 5 "YRS/g/01coloring.hpp"

TE vc<int> coloring01(graph<T, 0> &g) {
  assert(g.isok());
  int N = g.N;
  dsu fa(N << 1);
  for (Z &&e : g.es) fa.merge(e.f, e.to + N), fa.merge(e.to, e.f + N);
  vc<int> c(N << 1, -1);
  FOR(i, N) if (fa[i] == i and c[fa[i]] < 0) c[fa[i]] = 0, c[fa[i + N]] = 1;
  FOR(i, N) if (fa[i] == fa[i + N]) return {};
  FOR(i, N) c[i] = c[fa[i]];
  return c.resize(N), c;
}
#line 2 "YRS/g/scc.hpp"

#line 4 "YRS/g/scc.hpp"

TE pair<int, vc<int>> scc(graph<T, 1> &g) {
  assert(g.isok());
  int N = g.N, t = 0, c = 0;
  vc<int> dfn(N), low(N), id(N), s;

  Z f = [&](Z &f, int n) -> void {
    dfn[n] = low[n] = ++t;
    s.ep(n);
    for (Z &&e : g[n]) {
      if (dfn[e.to]) chmin(low[n], dfn[e.to]);
      else f(f, e.to), chmin(low[n], low[e.to]);
    }
    if (dfn[n] == low[n]) {
      int x = pop(s);
      for (; x != n; x = pop(s)) id[x] = c, dfn[x] = N;
      id[x] = c++, dfn[x] = N;
    }
  };
  FOR(i, N) if (not dfn[i]) f(f, i);
  FOR(i, N) id[i] = c - id[i] - 1; // del
  return {c, id};
}

vc<vc<int>> get_scc_group(int c, const vc<int> &id) {
  vc<vc<int>> scc(c);
  int N = len(id);
  vc<int> sz(c); // del
  FOR(i, N) ++sz[id[i]]; // del
  FOR(i, c) scc[i].reserve(sz[i]); // del
  FOR(i, N) scc[id[i]].ep(i);
  return scc;
}

TE graph<int, 1> scc_dag(graph<T, 1> &g, int c, const vc<int> &id) {
  assert(g.isok());
  graph<int, 1> ng(c);
  vc<ull> es;
  for (Z &&e : g.es) {
    int x = id[e.f], y = id[e.to];
    if (x == y) continue;
    es.ep(ull(x) << 32 | y);
  }
  unique(es);
  ng.es.reserve(len(es)); // del
  for (ull s : es) ng.add(s >> 32, s & uint(-1));
  return ng.build(), ng;
}
#line 2 "YRS/ds/basic/queue.hpp"

// queue_vector_ver
TE struct myqueue {
  int l;
  vc<T> q;

  myqueue() : l(0) {}
  myqueue(int N) : l(0) { q.reserve(N); }
  myqueue(const vc<T> &q) : l(0), q(q) {}

  T &operator[](int x) { 
    if (x < 0) return q.end()[x];
    return q[l + x]; 
  }

  const T &operator[](int x) const { 
    if (x < 0) return q.end()[x];
    return q[l + x]; 
  }

  T &front() { return q[l]; }

  const T &front() const { return q[l]; }

  T &back() { return q.back(); }

  const T &back() const { return q.back(); }

  int size() const { return len(q) - l; }

  bool empty() const { return l == len(q); }

  T pop() { return q[l++]; }

  void push(const T &x) { push_back(x); }

  void push_back(const T& v) { q.push_back(v); }

  void pop_back() { q.pop_back(); }
  
  void clear() { q.clear(), l = 0; }

  vc<T>::iterator end() { return q.end(); }

  template <typename... S>
  void emplace(S &&...args) { q.ep(std::forward<S>(args)...); }
};
#define queue myqueue

TE T pop(queue<T> &q) {
  T x = q.front();
  return q.pop(), x;
}
#line 7 "YRS/flow/BM.hpp"

TE struct B_matching {
  using G = graph<T, 0>;
  int N;
  G &g;
  vc<int> col, dis, mat;
  vc<u8> vis;

  B_matching(G &g) : N(g.N), g(g), dis(N, -1), mat(N, -1) {
    col = coloring01(g);
    if (N > 0) assert(not col.empty());
    while (1) {
      bfs();
      vis.assign(N, 0);
      int f = 0;
      FOR(i, N) if (not col[i] and mat[i] == -1 and dfs(i)) ++f;
      if (not f) break;
    }
  }
  B_matching(G &g, const vc<int> &col)
      : N(g.N), g(g), col(col), dis(N, -1), mat(N, -1) {
    if (N > 0) assert(not col.empty());
    while (1) {
      bfs();
      fill(all(vis), 0);
      int f = 0;
      FOR(i, N) if (not col[i] and mat[i] == -1 and dfs(i)) ++f;
      if (not f) break;
    }
  }

  void bfs() {
    fill(all(dis), -1);
    queue<int> q;
    FOR(i, N) if (not col[i] and mat[i] == -1) q.eb(i), dis[i] = 0;
    while (not q.empty()) {
      int n = pop(q);
      for (Z &&e : g[n]) {
        dis[e.to] = 0;
        int w = mat[e.to];
        if (w != -1 and dis[w] == -1) dis[w] = dis[n] + 1, q.eb(w);
      }
    }
  }
  bool dfs(int n) {
    vis[n] = 1;
    for (Z &&e : g[n]) {
      int w = mat[e.to];
      if (w == -1 or (not vis[w] and dis[w] == dis[n] + 1 and dfs(w))) {
        mat[e.to] = n, mat[n] = e.to;
        return 1;
      }
    }
    return 0;
  }
  vc<PII> matching() {
    vc<PII> s;
    FOR(i, N) if (i < mat[i]) s.ep(i, mat[i]);
    return s;
  }

  // https://www.luogu.com.cn/problem/P7368
  // 最小点覆盖
  vc<int> vcov() {
    vc<int> s;
    FOR(i, N) if (col[i] ^ (dis[i] == -1)) s.ep(i);
    return s;
  }
  // https://www.luogu.com.cn/problem/P3355
  // 最大独立集
  vc<int> iset() {
    vc<int> s;
    FOR(i, N) if (not(col[i] ^ (dis[i] == -1))) s.ep(i);
    return s;
  }
  // 最小边覆盖
  vc<int> ecov() {
    vc<u8> vis(N);
    vc<int> s;
    FOR(i, N) for (Z &&e : g[i]) {
      if (vis[e.f] or vis[e.to]) continue;
      if (mat[e.f] == e.to) s.ep(e.id), vis[e.f] = vis[e.to] = 1;
    }
    FOR(i, N) for (Z &&e : g[i]) {
      if (not vis[e.f]) s.ep(e.id), vis[e.f] = 1;
      if (not vis[e.to]) s.ep(e.to), vis[e.to] = 1;
    }
    return sort(s), s;
  }
  
  // https://yukicoder.me/problems/no/1745
  // Dulmage–Mendelsohn 分解
  // - 可以作为最大匹配的条件：具有相同的 W
  // - 边 x-y 必定被使用：具有相同 W 的边唯一
  // - 从 col=0 到 col=1 的边：W[l] <= W[r]
  // - col=0 的点必定被使用：W=1,2,...,K
  // - col=1 的点必定被使用：W=0,1,...,K-1
  pair<int, vc<int>> DM_decomposition() {
    // 从非饱和点开始的搜索
    vc<int> w(N, -1), q;
    Z add = [&](int n, int x) -> void {
      if (w[n] == -1) {
        w[n] = x;
        q.ep(n);
      }
    };
    FOR(i, N) if (mat[i] == -1 and col[i] == 0) add(i, 0);
    FOR(i, N) if (mat[i] == -1 and col[i] == 1) add(i, inf<int>);

    while (not q.empty()) {
      int n = pop(q);
      if (mat[n] != -1) add(mat[n], w[n]);
      if (col[n] == 0 and w[n] == 0) {
        for (Z &&e : g[n]) add(e.to, w[n]);
      }
      if (col[n] == 1 and w[n] == inf<int>) {
        for (Z &&e : g[n]) add(e.to, w[n]);
      }
    }

    // 从剩余的点构成的图中进行强连通分量分解
    vc<int> v;
    FOR(i, N) if (w[i] == -1) v.ep(i);
    int n = len(v);
    graph<int, 1> ng(n);
    FOR(i, n) {
      int x = v[i];
      if (mat[x] != -1) ng.add(i, lb(v, mat[x]));
      if (col[x] == 0) {
        for (Z &&e : g[x]) {
          if (w[e.to] != -1 or e.to == mat[x]) continue;
          ng.add(i, lb(v, e.to));
        }
      }
    }
    ng.build();
    Z [c, id] = scc(ng);
    ++c;
    FOR(i, n) w[v[i]] = id[i] + 1;
    FOR(i, N) if (w[i] == inf<int>) w[i] = c;
    return {c, w};
  }
};
#line 11 "No_179_\u5857\u308a\u5206\u3051.cpp"

void Yorisou() {
  INT(N, M);
  VEC(string, a, N);
  int s = 0, t = 0;
  retsu<int> id(N, M);
  FOR(i, N) FOR(k, M) {
    if (a[i][k] == '#') ++s, id[i][k] = t++;
  }
  if (s == 0) return NO();

  Z ck = [&](int x, int y) {
    return x >= 0 and x < N and y >= 0 and y < M and a[x][y] == '#';
  };
  FOR(x, -N, N) FOR(y, -M, M) if (x != 0 or y != 0) {
    graph g(s);
    FOR(i, N) FOR(k, M) if (a[i][k] == '#' and ck(i + x, k + y)) {
      g.add(id[i][k], id[i + x][k + y]);
    }
    g.build();
    if (2 * len(B_matching(g).matching()) == s) return YES();
  }
  NO();
}
constexpr int tests = 0, fl = 0, DB = 10;
#line 1 "YRS/aa/main.hpp"
int main() {
  cin.tie(0)->sync_with_stdio(0);
  int T = 1;
  if (fl) cerr.tie(0);
  if (tests and not fl) IN(T);
  for (int i = 0; i < T or fl; ++i) {
    Yorisou();
    if (fl and i % DB == 0) cerr << "Case: " << i << '\n';
  }
  return 0;
}
#line 37 "No_179_\u5857\u308a\u5206\u3051.cpp"