#line 1 "library/my_template.hpp"
#if defined(LOCAL)
#include <my_template_compiled.hpp>
#else
#pragma GCC optimize("Ofast")
#pragma GCC optimize("unroll-loops")

#include <bits/stdc++.h>

using namespace std;

using ll = long long;
using u32 = unsigned int;
using u64 = unsigned long long;
using i128 = __int128;

template <class T>
constexpr T infty = 0;
template <>
constexpr int infty<int> = 1'000'000'000;
template <>
constexpr ll infty<ll> = ll(infty<int>) * infty<int> * 2;
template <>
constexpr u32 infty<u32> = infty<int>;
template <>
constexpr u64 infty<u64> = infty<ll>;
template <>
constexpr i128 infty<i128> = i128(infty<ll>) * infty<ll>;
template <>
constexpr double infty<double> = infty<ll>;
template <>
constexpr long double infty<long double> = infty<ll>;

using pi = pair<ll, ll>;
using vi = vector<ll>;
template <class T>
using vc = vector<T>;
template <class T>
using vvc = vector<vc<T>>;
template <class T>
using vvvc = vector<vvc<T>>;
template <class T>
using vvvvc = vector<vvvc<T>>;
template <class T>
using vvvvvc = vector<vvvvc<T>>;
template <class T>
using pq = priority_queue<T>;
template <class T>
using pqg = priority_queue<T, vector<T>, greater<T>>;

#define vv(type, name, h, ...) \
  vector<vector<type>> name(h, vector<type>(__VA_ARGS__))
#define vvv(type, name, h, w, ...)   \
  vector<vector<vector<type>>> name( \
      h, vector<vector<type>>(w, vector<type>(__VA_ARGS__)))
#define vvvv(type, name, a, b, c, ...)       \
  vector<vector<vector<vector<type>>>> name( \
      a, vector<vector<vector<type>>>(       \
             b, vector<vector<type>>(c, vector<type>(__VA_ARGS__))))

// https://trap.jp/post/1224/
#define FOR1(a) for (ll _ = 0; _ < ll(a); ++_)
#define FOR2(i, a) for (ll i = 0; i < ll(a); ++i)
#define FOR3(i, a, b) for (ll i = a; i < ll(b); ++i)
#define FOR4(i, a, b, c) for (ll i = a; i < ll(b); i += (c))
#define FOR1_R(a) for (ll i = (a)-1; i >= ll(0); --i)
#define FOR2_R(i, a) for (ll i = (a)-1; i >= ll(0); --i)
#define FOR3_R(i, a, b) for (ll i = (b)-1; i >= ll(a); --i)
#define overload4(a, b, c, d, e, ...) e
#define overload3(a, b, c, d, ...) d
#define FOR(...) overload4(__VA_ARGS__, FOR4, FOR3, FOR2, FOR1)(__VA_ARGS__)
#define FOR_R(...) overload3(__VA_ARGS__, FOR3_R, FOR2_R, FOR1_R)(__VA_ARGS__)

#define FOR_subset(t, s) \
  for (ll t = (s); t >= 0; t = (t == 0 ? -1 : (t - 1) & (s)))
#define all(x) x.begin(), x.end()
#define len(x) ll(x.size())
#define elif else if

#define eb emplace_back
#define mp make_pair
#define mt make_tuple
#define fi first
#define se second

#define stoi stoll

int popcnt(int x) { return __builtin_popcount(x); }
int popcnt(u32 x) { return __builtin_popcount(x); }
int popcnt(ll x) { return __builtin_popcountll(x); }
int popcnt(u64 x) { return __builtin_popcountll(x); }
// (0, 1, 2, 3, 4) -> (-1, 0, 1, 1, 2)
int topbit(int x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); }
int topbit(u32 x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); }
int topbit(ll x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); }
int topbit(u64 x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); }
// (0, 1, 2, 3, 4) -> (-1, 0, 1, 0, 2)
int lowbit(int x) { return (x == 0 ? -1 : __builtin_ctz(x)); }
int lowbit(u32 x) { return (x == 0 ? -1 : __builtin_ctz(x)); }
int lowbit(ll x) { return (x == 0 ? -1 : __builtin_ctzll(x)); }
int lowbit(u64 x) { return (x == 0 ? -1 : __builtin_ctzll(x)); }

template <typename T, typename U>
T ceil(T x, U y) {
  return (x > 0 ? (x + y - 1) / y : x / y);
}
template <typename T, typename U>
T floor(T x, U y) {
  return (x > 0 ? x / y : (x - y + 1) / y);
}
template <typename T, typename U>
pair<T, T> divmod(T x, U y) {
  T q = floor(x, y);
  return {q, x - q * y};
}

template <typename T, typename U>
T SUM(const vector<U> &A) {
  T sum = 0;
  for (auto &&a: A) sum += a;
  return sum;
}

#define MIN(v) *min_element(all(v))
#define MAX(v) *max_element(all(v))
#define LB(c, x) distance((c).begin(), lower_bound(all(c), (x)))
#define UB(c, x) distance((c).begin(), upper_bound(all(c), (x)))
#define UNIQUE(x) \
  sort(all(x)), x.erase(unique(all(x)), x.end()), x.shrink_to_fit()

template <typename T>
T POP(deque<T> &que) {
  T a = que.front();
  que.pop_front();
  return a;
}
template <typename T>
T POP(pq<T> &que) {
  T a = que.top();
  que.pop();
  return a;
}
template <typename T>
T POP(pqg<T> &que) {
  assert(!que.empty());
  T a = que.top();
  que.pop();
  return a;
}
template <typename T>
T POP(vc<T> &que) {
  assert(!que.empty());
  T a = que.back();
  que.pop_back();
  return a;
}

template <typename F>
ll binary_search(F check, ll ok, ll ng, bool check_ok = true) {
  if (check_ok) assert(check(ok));
  while (abs(ok - ng) > 1) {
    auto x = (ng + ok) / 2;
    tie(ok, ng) = (check(x) ? mp(x, ng) : mp(ok, x));
  }
  return ok;
}
template <typename F>
double binary_search_real(F check, double ok, double ng, int iter = 100) {
  FOR(iter) {
    double x = (ok + ng) / 2;
    tie(ok, ng) = (check(x) ? mp(x, ng) : mp(ok, x));
  }
  return (ok + ng) / 2;
}

template <class T, class S>
inline bool chmax(T &a, const S &b) {
  return (a < b ? a = b, 1 : 0);
}
template <class T, class S>
inline bool chmin(T &a, const S &b) {
  return (a > b ? a = b, 1 : 0);
}

// ? は -1
vc<int> s_to_vi(const string &S, char first_char) {
  vc<int> A(S.size());
  FOR(i, S.size()) { A[i] = (S[i] != '?' ? S[i] - first_char : -1); }
  return A;
}

template <typename T, typename U>
vector<T> cumsum(vector<U> &A, int off = 1) {
  int N = A.size();
  vector<T> B(N + 1);
  FOR(i, N) { B[i + 1] = B[i] + A[i]; }
  if (off == 0) B.erase(B.begin());
  return B;
}

// stable sort
template <typename T>
vector<int> argsort(const vector<T> &A) {
  vector<int> ids(len(A));
  iota(all(ids), 0);
  sort(all(ids),
       [&](int i, int j) { return (A[i] == A[j] ? i < j : A[i] < A[j]); });
  return ids;
}

// A[I[0]], A[I[1]], ...
template <typename T>
vc<T> rearrange(const vc<T> &A, const vc<int> &I) {
  vc<T> B(len(I));
  FOR(i, len(I)) B[i] = A[I[i]];
  return B;
}
#endif
#line 1 "library/other/io.hpp"
// based on yosupo's fastio
#include <unistd.h>

namespace fastio {
#define FASTIO
// クラスが read(), print() を持っているかを判定するメタ関数
struct has_write_impl {
  template <class T>
  static auto check(T &&x) -> decltype(x.write(), std::true_type{});

  template <class T>
  static auto check(...) -> std::false_type;
};

template <class T>
class has_write : public decltype(has_write_impl::check<T>(std::declval<T>())) {
};

struct has_read_impl {
  template <class T>
  static auto check(T &&x) -> decltype(x.read(), std::true_type{});

  template <class T>
  static auto check(...) -> std::false_type;
};

template <class T>
class has_read : public decltype(has_read_impl::check<T>(std::declval<T>())) {};

struct Scanner {
  FILE *fp;
  char line[(1 << 15) + 1];
  size_t st = 0, ed = 0;
  void reread() {
    memmove(line, line + st, ed - st);
    ed -= st;
    st = 0;
    ed += fread(line + ed, 1, (1 << 15) - ed, fp);
    line[ed] = '\0';
  }
  bool succ() {
    while (true) {
      if (st == ed) {
        reread();
        if (st == ed) return false;
      }
      while (st != ed && isspace(line[st])) st++;
      if (st != ed) break;
    }
    if (ed - st <= 50) {
      bool sep = false;
      for (size_t i = st; i < ed; i++) {
        if (isspace(line[i])) {
          sep = true;
          break;
        }
      }
      if (!sep) reread();
    }
    return true;
  }
  template <class T, enable_if_t<is_same<T, string>::value, int> = 0>
  bool read_single(T &ref) {
    if (!succ()) return false;
    while (true) {
      size_t sz = 0;
      while (st + sz < ed && !isspace(line[st + sz])) sz++;
      ref.append(line + st, sz);
      st += sz;
      if (!sz || st != ed) break;
      reread();
    }
    return true;
  }
  template <class T, enable_if_t<is_integral<T>::value, int> = 0>
  bool read_single(T &ref) {
    if (!succ()) return false;
    bool neg = false;
    if (line[st] == '-') {
      neg = true;
      st++;
    }
    ref = T(0);
    while (isdigit(line[st])) { ref = 10 * ref + (line[st++] & 0xf); }
    if (neg) ref = -ref;
    return true;
  }
  template <typename T,
            typename enable_if<has_read<T>::value>::type * = nullptr>
  inline bool read_single(T &x) {
    x.read();
    return true;
  }
  bool read_single(double &ref) {
    string s;
    if (!read_single(s)) return false;
    ref = std::stod(s);
    return true;
  }
  bool read_single(char &ref) {
    string s;
    if (!read_single(s) || s.size() != 1) return false;
    ref = s[0];
    return true;
  }
  template <class T>
  bool read_single(vector<T> &ref) {
    for (auto &d: ref) {
      if (!read_single(d)) return false;
    }
    return true;
  }
  template <class T, class U>
  bool read_single(pair<T, U> &p) {
    return (read_single(p.first) && read_single(p.second));
  }
  template <size_t N = 0, typename T>
  void read_single_tuple(T &t) {
    if constexpr (N < std::tuple_size<T>::value) {
      auto &x = std::get<N>(t);
      read_single(x);
      read_single_tuple<N + 1>(t);
    }
  }
  template <class... T>
  bool read_single(tuple<T...> &tpl) {
    read_single_tuple(tpl);
    return true;
  }
  void read() {}
  template <class H, class... T>
  void read(H &h, T &... t) {
    bool f = read_single(h);
    assert(f);
    read(t...);
  }
  Scanner(FILE *fp) : fp(fp) {}
};

struct Printer {
  Printer(FILE *_fp) : fp(_fp) {}
  ~Printer() { flush(); }

  static constexpr size_t SIZE = 1 << 15;
  FILE *fp;
  char line[SIZE], small[50];
  size_t pos = 0;
  void flush() {
    fwrite(line, 1, pos, fp);
    pos = 0;
  }
  void write(const char val) {
    if (pos == SIZE) flush();
    line[pos++] = val;
  }
  template <class T, enable_if_t<is_integral<T>::value, int> = 0>
  void write(T val) {
    if (pos > (1 << 15) - 50) flush();
    if (val == 0) {
      write('0');
      return;
    }
    if (val < 0) {
      write('-');
      val = -val; // todo min
    }
    size_t len = 0;
    while (val) {
      small[len++] = char(0x30 | (val % 10));
      val /= 10;
    }
    for (size_t i = 0; i < len; i++) { line[pos + i] = small[len - 1 - i]; }
    pos += len;
  }
  void write(const string s) {
    for (char c: s) write(c);
  }
  void write(const char *s) {
    size_t len = strlen(s);
    for (size_t i = 0; i < len; i++) write(s[i]);
  }
  void write(const double x) {
    ostringstream oss;
    oss << fixed << setprecision(15) << x;
    string s = oss.str();
    write(s);
  }
  void write(const long double x) {
    ostringstream oss;
    oss << fixed << setprecision(15) << x;
    string s = oss.str();
    write(s);
  }
  template <typename T,
            typename enable_if<has_write<T>::value>::type * = nullptr>
  inline void write(T x) {
    x.write();
  }
  template <class T>
  void write(const vector<T> val) {
    auto n = val.size();
    for (size_t i = 0; i < n; i++) {
      if (i) write(' ');
      write(val[i]);
    }
  }
  template <class T, class U>
  void write(const pair<T, U> val) {
    write(val.first);
    write(' ');
    write(val.second);
  }
  template <size_t N = 0, typename T>
  void write_tuple(const T t) {
    if constexpr (N < std::tuple_size<T>::value) {
      if constexpr (N > 0) { write(' '); }
      const auto x = std::get<N>(t);
      write(x);
      write_tuple<N + 1>(t);
    }
  }
  template <class... T>
  bool write(tuple<T...> tpl) {
    write_tuple(tpl);
    return true;
  }
  template <class T, size_t S>
  void write(const array<T, S> val) {
    auto n = val.size();
    for (size_t i = 0; i < n; i++) {
      if (i) write(' ');
      write(val[i]);
    }
  }
  void write(i128 val) {
    string s;
    bool negative = 0;
    if (val < 0) {
      negative = 1;
      val = -val;
    }
    while (val) {
      s += '0' + int(val % 10);
      val /= 10;
    }
    if (negative) s += "-";
    reverse(all(s));
    if (len(s) == 0) s = "0";
    write(s);
  }
};
Scanner scanner = Scanner(stdin);
Printer printer = Printer(stdout);
void flush() { printer.flush(); }
void print() { printer.write('\n'); }
template <class Head, class... Tail>
void print(Head &&head, Tail &&... tail) {
  printer.write(head);
  if (sizeof...(Tail)) printer.write(' ');
  print(forward<Tail>(tail)...);
}

void read() {}
template <class Head, class... Tail>
void read(Head &head, Tail &... tail) {
  scanner.read(head);
  read(tail...);
}
} // namespace fastio
using fastio::print;
using fastio::flush;
using fastio::read;

#define INT(...)   \
  int __VA_ARGS__; \
  read(__VA_ARGS__)
#define LL(...)   \
  ll __VA_ARGS__; \
  read(__VA_ARGS__)
#define STR(...)      \
  string __VA_ARGS__; \
  read(__VA_ARGS__)
#define CHAR(...)   \
  char __VA_ARGS__; \
  read(__VA_ARGS__)
#define DBL(...)      \
  double __VA_ARGS__; \
  read(__VA_ARGS__)

#define VEC(type, name, size) \
  vector<type> name(size);    \
  read(name)
#define VV(type, name, h, w)                     \
  vector<vector<type>> name(h, vector<type>(w)); \
  read(name)

void YES(bool t = 1) { print(t ? "YES" : "NO"); }
void NO(bool t = 1) { YES(!t); }
void Yes(bool t = 1) { print(t ? "Yes" : "No"); }
void No(bool t = 1) { Yes(!t); }
void yes(bool t = 1) { print(t ? "yes" : "no"); }
void no(bool t = 1) { yes(!t); }
#line 2 "library/graph/base.hpp"

template <typename T>
struct Edge {
  int frm, to;
  T cost;
  int id;
};

template <typename T = int, bool directed = false>
struct Graph {
  int N, M;
  using cost_type = T;
  using edge_type = Edge<T>;
  vector<edge_type> edges;
  vector<int> indptr;
  vector<edge_type> csr_edges;
  vc<int> vc_deg, vc_indeg, vc_outdeg;
  bool prepared;

  class OutgoingEdges {
  public:
    OutgoingEdges(const Graph* G, int l, int r) : G(G), l(l), r(r) {}

    const edge_type* begin() const {
      if (l == r) { return 0; }
      return &G->csr_edges[l];
    }

    const edge_type* end() const {
      if (l == r) { return 0; }
      return &G->csr_edges[r];
    }

  private:
    const Graph* G;
    int l, r;
  };

  bool is_prepared() { return prepared; }
  constexpr bool is_directed() { return directed; }

  Graph() : N(0), M(0), prepared(0) {}
  Graph(int N) : N(N), M(0), prepared(0) {}

  void build(int n) {
    N = n, M = 0;
    prepared = 0;
    edges.clear();
    indptr.clear();
    csr_edges.clear();
    vc_deg.clear();
    vc_indeg.clear();
    vc_outdeg.clear();
  }

  void add(int frm, int to, T cost = 1, int i = -1) {
    assert(!prepared);
    assert(0 <= frm && 0 <= to && to < N);
    if (i == -1) i = M;
    auto e = edge_type({frm, to, cost, i});
    edges.eb(e);
    ++M;
  }

  // wt, off
  void read_tree(bool wt = false, int off = 1) { read_graph(N - 1, wt, off); }

  void read_graph(int M, bool wt = false, int off = 1) {
    for (int m = 0; m < M; ++m) {
      INT(a, b);
      a -= off, b -= off;
      if (!wt) {
        add(a, b);
      } else {
        T c;
        read(c);
        add(a, b, c);
      }
    }
    build();
  }

  void read_parent(int off = 1) {
    for (int v = 1; v < N; ++v) {
      INT(p);
      p -= off;
      add(p, v);
    }
    build();
  }

  void build() {
    assert(!prepared);
    prepared = true;
    indptr.assign(N + 1, 0);
    for (auto&& e: edges) {
      indptr[e.frm + 1]++;
      if (!directed) indptr[e.to + 1]++;
    }
    for (int v = 0; v < N; ++v) { indptr[v + 1] += indptr[v]; }
    auto counter = indptr;
    csr_edges.resize(indptr.back() + 1);
    for (auto&& e: edges) {
      csr_edges[counter[e.frm]++] = e;
      if (!directed)
        csr_edges[counter[e.to]++] = edge_type({e.to, e.frm, e.cost, e.id});
    }
  }

  OutgoingEdges operator[](int v) const {
    assert(prepared);
    return {this, indptr[v], indptr[v + 1]};
  }

  vc<int> deg_array() {
    if (vc_deg.empty()) calc_deg();
    return vc_deg;
  }

  pair<vc<int>, vc<int>> deg_array_inout() {
    if (vc_indeg.empty()) calc_deg_inout();
    return {vc_indeg, vc_outdeg};
  }

  int deg(int v) {
    if (vc_deg.empty()) calc_deg();
    return vc_deg[v];
  }

  int in_deg(int v) {
    if (vc_indeg.empty()) calc_deg_inout();
    return vc_indeg[v];
  }

  int out_deg(int v) {
    if (vc_outdeg.empty()) calc_deg_inout();
    return vc_outdeg[v];
  }

  void debug() {
    print("Graph");
    if (!prepared) {
      print("frm to cost id");
      for (auto&& e: edges) print(e.frm, e.to, e.cost, e.id);
    } else {
      print("indptr", indptr);
      print("frm to cost id");
      FOR(v, N) for (auto&& e: (*this)[v]) print(e.frm, e.to, e.cost, e.id);
    }
  }

  // G における頂点 V[i] が、新しいグラフで i になるようにする
  Graph<T, directed> rearrange(vc<int> V) {
    int n = len(V);
    map<int, int> MP;
    FOR(i, n) MP[V[i]] = i;
    Graph<T, directed> G(n);
    for (auto&& e: edges) {
      if (MP.count(e.frm) && MP.count(e.to)) {
        G.add(MP[e.frm], MP[e.to], e.cost);
      }
    }
    G.build();
    return G;
  }

private:
  void calc_deg() {
    assert(vc_deg.empty());
    vc_deg.resize(N);
    for (auto&& e: edges) vc_deg[e.frm]++, vc_deg[e.to]++;
  }

  void calc_deg_inout() {
    assert(vc_indeg.empty());
    vc_indeg.resize(N);
    vc_outdeg.resize(N);
    for (auto&& e: edges) { vc_indeg[e.to]++, vc_outdeg[e.frm]++; }
  }
};
#line 2 "library/graph/bipartite_vertex_coloring.hpp"

#line 2 "library/ds/unionfind/unionfind.hpp"

struct UnionFind {
  int n, n_comp;
  vc<int> dat; // par or (-size)
  UnionFind(int n = 0) { build(n); }

  void build(int m) {
    n = m, n_comp = m;
    dat.assign(n, -1);
  }

  void reset() { build(n); }

  int operator[](int x) {
    while (dat[x] >= 0) {
      int pp = dat[dat[x]];
      if (pp < 0) { return dat[x]; }
      x = dat[x] = pp;
    }
    return x;
  }

  ll size(int x) {
    assert(dat[x] < 0);
    return -dat[x];
  }

  bool merge(int x, int y) {
    x = (*this)[x], y = (*this)[y];
    if (x == y) return false;
    if (-dat[x] < -dat[y]) swap(x, y);
    dat[x] += dat[y], dat[y] = x, n_comp--;
    return true;
  }
};
#line 5 "library/graph/bipartite_vertex_coloring.hpp"

// 二部グラフでなかった場合には empty
template <typename Graph>
vc<int> bipartite_vertex_coloring(Graph& G) {
  assert(G.is_prepared());

  int n = G.N;
  UnionFind uf(2 * n);
  for (auto&& e: G.edges) {
    int u = e.frm, v = e.to;
    if (e.cost == 0) uf.merge(u, v), uf.merge(u + n, v + n);
    if (e.cost != 0) uf.merge(u + n, v), uf.merge(u, v + n);
  }

  vc<int> color(2 * n, -1);
  FOR(v, n) if (uf[v] == v && color[uf[v]] < 0) {
    color[uf[v]] = 0;
    color[uf[v + n]] = 1;
  }
  FOR(v, n) color[v] = color[uf[v]];
  color.resize(n);
  FOR(v, n) if (uf[v] == uf[v + n]) return {};
  return color;
}
#line 3 "library/graph/strongly_connected_component.hpp"

template <typename Graph>
pair<int, vc<int>> strongly_connected_component(Graph& G) {
  assert(G.is_directed());
  assert(G.is_prepared());
  int N = G.N;
  int C = 0;
  vc<int> comp(N);
  vc<int> low(N);
  vc<int> ord(N, -1);
  vc<int> visited;
  int now = 0;

  auto dfs = [&](auto self, int v) -> void {
    low[v] = now;
    ord[v] = now;
    ++now;
    visited.eb(v);
    for (auto&& [frm, to, cost, id]: G[v]) {
      if (ord[to] == -1) {
        self(self, to);
        chmin(low[v], low[to]);
      } else {
        chmin(low[v], ord[to]);
      }
    }
    if (low[v] == ord[v]) {
      while (1) {
        int u = visited.back();
        visited.pop_back();
        ord[u] = N;
        comp[u] = C;
        if (u == v) break;
      }
      ++C;
    }
  };
  FOR(v, N) {
    if (ord[v] == -1) dfs(dfs, v);
  }
  FOR(v, N) comp[v] = C - 1 - comp[v];
  return {C, comp};
}

template <typename GT>
Graph<int, 1> scc_dag(GT& G, int C, vc<int>& comp) {
  Graph<int, 1> DAG(C);
  vvc<int> edges(C);
  for (auto&& e: G.edges) {
    int x = comp[e.frm], y = comp[e.to];
    if (x == y) continue;
    edges[x].eb(y);
  }
  FOR(c, C) {
    UNIQUE(edges[c]);
    for (auto&& to: edges[c]) DAG.add(c, to);
  }
  DAG.build();
  return DAG;
}
#line 4 "library/flow/bipartite.hpp"

template <typename GT>
struct BipartiteMatching {
  int N;
  GT& G;
  vc<int> color;
  vc<int> dist, match;
  vc<int> vis;

  BipartiteMatching(GT& G) : N(G.N), G(G), dist(G.N, -1), match(G.N, -1) {
    color = bipartite_vertex_coloring(G);
    assert(!color.empty());
    while (1) {
      bfs();
      vis.assign(N, false);
      int flow = 0;
      FOR(v, N) if (!color[v] && match[v] == -1 && dfs(v))++ flow;
      if (!flow) break;
    }
  }

  BipartiteMatching(GT& G, vc<int> color)
      : N(G.N), G(G), color(color), dist(G.N, -1), match(G.N, -1) {
    while (1) {
      bfs();
      vis.assign(N, false);
      int flow = 0;
      FOR(v, N) if (!color[v] && match[v] == -1 && dfs(v))++ flow;
      if (!flow) break;
    }
  }

  void bfs() {
    dist.assign(N, -1);
    queue<int> que;
    FOR(v, N) if (!color[v] && match[v] == -1) que.emplace(v), dist[v] = 0;
    while (!que.empty()) {
      int v = que.front();
      que.pop();
      for (auto&& e: G[v]) {
        dist[e.to] = 0;
        int w = match[e.to];
        if (w != -1 && dist[w] == -1) dist[w] = dist[v] + 1, que.emplace(w);
      }
    }
  }

  bool dfs(int v) {
    vis[v] = 1;
    for (auto&& e: G[v]) {
      int w = match[e.to];
      if (w == -1 || (!vis[w] && dist[w] == dist[v] + 1 && dfs(w))) {
        match[e.to] = v, match[v] = e.to;
        return true;
      }
    }
    return false;
  }

  vc<pair<int, int>> matching() {
    vc<pair<int, int>> res;
    FOR(v, N) if (v < match[v]) res.eb(v, match[v]);
    return res;
  }

  vc<int> vertex_cover() {
    vc<int> res;
    FOR(v, N) if (color[v] ^ (dist[v] == -1)) { res.eb(v); }
    return res;
  }

  vc<int> independent_set() {
    vc<int> res;
    FOR(v, N) if (!(color[v] ^ (dist[v] == -1))) { res.eb(v); }
    return res;
  }

  vc<int> edge_cover() {
    vc<bool> done(N);
    vc<int> res;
    for (auto&& e: G.edges) {
      if (done[e.frm] || done[e.to]) continue;
      if (match[e.frm] == e.to) {
        res.eb(e.id);
        done[e.frm] = done[e.to] = 1;
      }
    }
    for (auto&& e: G.edges) {
      if (!done[e.frm]) {
        res.eb(e.id);
        done[e.frm] = 1;
      }
      if (!done[e.to]) {
        res.eb(e.id);
        done[e.to] = 1;
      }
    }
    sort(all(res));
    return res;
  }

  /* Dulmage–Mendelsohn decomposition
  https://en.wikipedia.org/wiki/Dulmage%E2%80%93Mendelsohn_decomposition
  http://www.misojiro.t.u-tokyo.ac.jp/~murota/lect-ouyousurigaku/dm050410.pdf
  https://hitonanode.github.io/cplib-cpp/graph/dulmage_mendelsohn_decomposition.hpp.html
  - 最大マッチングとしてありうる iff 同じ W を持つ
  - 辺 uv が必ず使われる：同じ W を持つ辺が唯一
  - color=0 から 1 への辺：W[l] <= W[r]
  - color=0 の点が必ず使われる：W=1,2,...,K
  - color=1 の点が必ず使われる：W=0,1,...,K-1
  */
  pair<int, vc<int>> DM_decomposition() {
    // 非飽和点からの探索
    vc<int> W(N, -1);
    vc<int> que;
    auto add = [&](int v, int x) -> void {
      if (W[v] == -1) {
        W[v] = x;
        que.eb(v);
      }
    };
    FOR(v, N) if (match[v] == -1 && color[v] == 0) add(v, 0);
    FOR(v, N) if (match[v] == -1 && color[v] == 1) add(v, infty<int>);
    while (len(que)) {
      auto v = POP(que);
      if (match[v] != -1) add(match[v], W[v]);
      if (color[v] == 0 && W[v] == 0) {
        for (auto&& e: G[v]) { add(e.to, W[v]); }
      }
      if (color[v] == 1 && W[v] == infty<int>) {
        for (auto&& e: G[v]) { add(e.to, W[v]); }
      }
    }
    // 残った点からなるグラフを作って強連結成分分解
    vc<int> V;
    FOR(v, N) if (W[v] == -1) V.eb(v);
    int n = len(V);
    Graph<bool, 1> DG(n);
    FOR(i, n) {
      int v = V[i];
      if (match[v] != -1) {
        int j = LB(V, match[v]);
        DG.add(i, j);
      }
      if (color[v] == 0) {
        for (auto&& e: G[v]) {
          if (W[e.to] != -1 || e.to == match[v]) continue;
          int j = LB(V, e.to);
          DG.add(i, j);
        }
      }
    }
    DG.build();
    auto [K, comp] = strongly_connected_component(DG);
    K += 1;
    // 答
    FOR(i, n) { W[V[i]] = 1 + comp[i]; }
    FOR(v, N) if (W[v] == infty<int>) W[v] = K;
    return {K, W};
  }

  void debug() {
    print("match", match);
    print("min vertex covor", vertex_cover());
    print("max indep set", independent_set());
    print("min edge cover", edge_cover());
  }
};
#line 4 "main.cpp"

void solve() {
  LL(N, M);
  ll H = 2 * N - 1, W = 2 * M - 1;
  VEC(string, S, H);

  auto isin = [&](ll x, ll y) -> bool {
    return (0 <= x && x < H && 0 <= y && y < W);
  };
  ll dx[] = {1, 1, 0, -1, -1, -1, 0, 1};
  ll dy[] = {0, 1, 1, 1, 0, -1, -1, -1};

  vv(int, A, N, M);
  FOR(i, N) FOR(j, M) {
    int x = S[2 * i][2 * j] - '0';
    int deg = 0;
    FOR(d, 8) if (isin(2 * i + dx[d], 2 * j + dy[d]))++ deg;
    A[i][j] = deg - x;
  }

  vv(int, outer, N, M);
  FOR(j, M) outer[0][j] = outer[N - 1][j] = 1;
  FOR(i, N) outer[i][0] = outer[i][M - 1] = 1;

  Graph<bool, 0> G(N * M);
  auto idx = [&](int i, int j) -> int { return M * i + j; };

  auto can = [&](int x1, int y1, int x2, int y2) -> bool {
    if (!isin(2 * x1, 2 * y1)) return 0;
    if (!isin(2 * x2, 2 * y2)) return 0;
    if (!A[x1][y1]) return 0;
    if (!A[x2][y2]) return 0;
    if (outer[x1][y1] != outer[x2][y2]) return 0;
    if (!outer[x1][y1]) return 1;
    // both outer
    int x = x1 + x2, y = y1 + y2;
    return ((x == 0) || (x == 2 * N - 2) || (y == 0) || (y == 2 * M - 2));
  };

  FOR(i, N) FOR(j, M) {
    if (can(i, j, i + 1, j)) G.add(idx(i, j), idx(i + 1, j));
    if (can(i, j, i, j + 1)) G.add(idx(i, j), idx(i, j + 1));
  }
  G.build();

  BipartiteMatching<decltype(G)> BM(G);
  auto match = BM.matching();
  for (auto&& [a, b]: match) {
    auto [x1, y1] = divmod(a, M);
    auto [x2, y2] = divmod(b, M);
    --A[x1][y1], --A[x2][y2];
  }
  FOR(i, N) FOR(j, M) if (A[i][j]) return NO();

  FOR(i, H) FOR(j, W) {
    if (i % 2 == 1 || j % 2 == 1) S[i][j] = '#';
  }
  for (auto&& [a, b]: match) {
    auto [x1, y1] = divmod(a, M);
    auto [x2, y2] = divmod(b, M);
    int x = x1 + x2, y = y1 + y2;
    S[x][y] = '.';
  }

  YES();
  FOR(i, H) print(S[i]);
}

signed main() {
  solve();
  return 0;
}