#include <bits/stdc++.h>
using namespace std;
using ll = long long;
using grid = vector<vector<char>>;
 
constexpr ll MOD = 1000000007;            
constexpr ll INF = 1050000000;             
constexpr ll LONGINF = 1050000000000000000; 
 
struct all_init {
  all_init() {
    cout.tie(nullptr);
    cin.tie(nullptr);
    ios::sync_with_stdio(false);
    cout << fixed << setprecision(11);
  };
} ALL_INIT;

struct edge {
 
  int from, to;
  ll cost;
  ll capa;
 
  edge(int s, int d) : from(s), to(d) {
    cost = 0;
    capa = 0;
  }
  edge(int s, int d, ll w) : from(s), to(d), cost(w) { capa = 0; }
  edge(int s, int d, ll x, ll y) : from(s), to(d), cost(x), capa(y) {}
 
  bool operator<(const edge &x) const { return cost < x.cost; }
};
using graph = vector<vector<edge>>;
 
#define CIN(vector_array_etc, n)         \
  for (int loop = 0; loop < n; loop++) { \
    cin >> vector_array_etc[loop];       \
  }
#define COUT(vector_array_etc, n)                                   \
  for (int LOOP = 0; LOOP < n; LOOP++) {                            \
    cout << vector_array_etc[LOOP] << (LOOP == n - 1 ? '\n' : ' '); \
  }
#define VC(Type_name) vector<Type_name>
#define SORT(vector_etc) sort(vector_etc.begin(), vector_etc.end())
#define ALL(vec_etc) vec_etc.begin(), vec_etc.end()
#define VCVC(Type_name) vector<vector<Type_name>>  
#define WARSHALL vector<vector<ll>> g(n, vector<ll>(n, LONGINF))
#define endl '\n'
 
template <class T>
bool chmax(T &a, const T &b) {
  if (a < b) {
    a = b;
    return true;
  }
  return false;
} 
template <class T>
bool chmin(T &a, const T &b) {
  if (b < a) {
    a = b;
    return true;
  }
  return false;
} 
template <typename T>
istream &operator>>(istream &is, vector<T> &Vec) {
  for (T &x : Vec) {
    is >> x;
  }
  return is;
}
template <typename V, typename H>
void resize(vector<V> &vec, const H head) {
  vec.resize(head);
}
template <typename V, typename H, typename... T>
void resize(vector<V> &vec, const H &head, const T... tail) {
  vec.resize(head);
  for (auto &v : vec) {
    resize(v, tail...);
  }
}
template <ll mod>
struct ModInt {
  long long val;
  constexpr ModInt(long long v = 0) noexcept : val(v % mod) {
    if (val < 0) v += mod;
  }
  constexpr int getmod() { return mod; }
  constexpr ModInt operator-() const noexcept { return val ? mod - val : 0; }
  constexpr ModInt operator+(const ModInt &r) const noexcept {
    return ModInt(*this) += r;
  }
  constexpr ModInt operator-(const ModInt &r) const noexcept {
    return ModInt(*this) -= r;
  }
  constexpr ModInt operator*(const ModInt &r) const noexcept {
    return ModInt(*this) *= r;
  }
  constexpr ModInt operator/(const ModInt &r) const noexcept {
    return ModInt(*this) /= r;
  }
  constexpr ModInt &operator+=(const ModInt &r) noexcept {
    val += r.val;
    if (val >= mod) val -= mod;
    return *this;
  }
  constexpr ModInt &operator-=(const ModInt &r) noexcept {
    val -= r.val;
    if (val < 0) val += mod;
    return *this;
  }
  constexpr ModInt &operator*=(const ModInt &r) noexcept {
    val = val * r.val % mod;
    return *this;
  }
  constexpr ModInt &operator/=(const ModInt &r) noexcept {
    long long a = r.val, b = mod, u = 1, v = 0;
    while (b) {
      long long t = a / b;
      a -= t * b;
      swap(a, b);
      u -= t * v;
      swap(u, v);
    }
    val = val * u % mod;
    if (val < 0) val += mod;
    return *this;
  }
  constexpr bool operator==(const ModInt &r) const noexcept {
    return this->val == r.val;
  }
  constexpr bool operator!=(const ModInt &r) const noexcept {
    return this->val != r.val;
  }
  friend ostream &operator<<(ostream &os, const ModInt<mod> &x) noexcept {
    return os << x.val;
  }
  friend istream &operator>>(istream &is, ModInt<mod> &x) noexcept {
    return is >> x.val;
  }
  friend constexpr ModInt<mod> modpow(const ModInt<mod> &a,
                                      long long n) noexcept {
    if (n == 0) return 1;
    auto t = modpow(a, n / 2);
    t = t * t;
    if (n & 1) t = t * a;
    return t;
  }
};
template <class T>
struct nCk {
  vector<T> fact_, inv_, finv_;
  constexpr nCk() {}
  constexpr nCk(int n) noexcept : fact_(n, 1), inv_(n, 1), finv_(n, 1) {
    init(n);
  }
  constexpr void init(int n) noexcept {
    fact_.assign(n, 1), inv_.assign(n, 1), finv_.assign(n, 1);
    ll MOD = 1000000007;
    for (ll i = 2; i < n; i++) {
      fact_[i] = fact_[i - 1] * i;
      inv_[i] = -inv_[MOD % i] * (MOD / i);
      finv_[i] = finv_[i - 1] * inv_[i];
    }
  }
  constexpr T com(ll n, ll k) const noexcept {
    if (n < k || n < 0 || k < 0) return 0;
    return fact_[n] * finv_[k] * finv_[n - k];
  }
  constexpr T fact(int n) const noexcept {
    if (n < 0) return 0;
    return fact_[n];
  }
  constexpr T inv(int n) const noexcept {
    if (n < 0) return 0;
    return inv_[n];
  }
  constexpr T finv(int n) const noexcept {
    if (n < 0) return 0;
    return finv_[n];
  }
};
 
int dx[] = {0, 1, -1, 0, 1, -1, 1, -1};  // i<4:4way i<8:8way
int dy[] = {1, 0, 0, -1, 1, -1, -1, 1};
 
ll PowMod(ll n, ll k, ll mod) {
  ll r = 1;
 
  for (; k > 0; k >>= 1) {
    if (k & 1) {
      r = (r * n) % mod;
    }
    n = (n * n) % mod;
  }
  return r;
}
ll Gcd(ll a, ll b) { 
  return b != 0 ? Gcd(b, a % b) : a;
}
ll Lcm(ll a, ll b) { 
  return a / Gcd(a, b) * b;
}
vector<string> Split(string s, string t) {
  vector<string> v;
  int p = s.find(t);
  if (p != s.npos) {
    v.push_back(s.substr(0, p));
    s = s.substr(p + (int)t.size());
  }
  v.push_back(s);
  return v;
}
vector<int> Lis(const vector<int> &a) {
//#define Index_of(as, x) distance(as.begin(), lower_bound(as.begin(), as.end(),
// x))
#define Index_of(as, x) \
  distance(as.begin(), upper_bound(as.begin(), as.end(), x))
  const int n = a.size();
  vector<int> A(n, INF);
  vector<int> id(n);
  for (int i = 0; i < n; ++i) {
    id[i] = Index_of(A, a[i]);
    A[id[i]] = a[i];
  }
  int m = *max_element(id.begin(), id.end());
  vector<int> b(m + 1);
  for (int i = n - 1; i >= 0; --i)
    if (id[i] == m) b[m--] = a[i];
  return b; 
}
string ReplaceString(string s, string target, string replacestring) {
  string::size_type Pos(s.find(target));
 
  while (Pos != string::npos) {
    s.replace(Pos, target.length(), replacestring);
    Pos = s.find(target, Pos + replacestring.length());
  }
 
  return s;
}
string LcsAlphabeticalMinOrder(string a, string b) {
  if (a.size() < b.size()) {
    swap(a, b);
  }
 
  int n = a.size(), m = b.size();
 
  vector<string> dp(m + 1);
 
  for (int i = 0; i < n; i++) {
    vector<string> to(m + 1);
    for (int j = 0; j < m; j++) {
      if (a[i] == b[j]) {
        to[j + 1] = dp[j] + a[i];
      } else {
        if (to[j].size() > dp[j + 1].size()) {
          to[j + 1] = to[j];
        } else if (to[j].size() < dp[j + 1].size()) {
          to[j + 1] = dp[j + 1];
        } else if (to[j] < dp[j + 1]) {
          to[j + 1] = to[j];
        } else {
          to[j + 1] = dp[j + 1];
        }
      }
    }
    dp.swap(to);
  }
  return dp[m];
}
string Lcs(const string &s, const string &t) {
  int dp[3001][3001];
  int n = s.size();
  int m = t.size();
  for (int i = 1; i <= n; i++) {
    for (int j = 1; j <= m; j++) {
      if (s[i - 1] == t[j - 1]) {
        dp[i][j] = dp[i - 1][j - 1] + 1;
      } else {
        dp[i][j] = max(dp[i - 1][j], dp[i][j - 1]);
      }
    }
  }
  string ans = "";
  int i = s.size(), j = t.size();
  while (i > 0 && j > 0) {
    if (s[i - 1] == t[j - 1]) {
      ans += s[i - 1];
      i--;
      j--;
    } else if (dp[i - 1][j] >= dp[i][j - 1])
      i--;
    else
      j--;
  }
  reverse(ans.begin(), ans.end());
  return ans;
}
vector<int> LcsInteger(const vector<int> &a, const vector<int> &b) {
#define index_of(as, x) \
  distance(as.begin(), lower_bound(as.begin(), as.end(), x))
  struct node {
    int value;
    node *next;
    node(int value, node *next) : value(value), next(next) {}
  };
  const int n = a.size(), m = b.size();
  map<int, vector<int>> M;
  for (int j = m - 1; j >= 0; --j) M[b[j]].push_back(j);
  vector<int> xs(n + 1, INF);
  xs[0] = -INF;
  vector<node *> link(n + 1);
  for (int i = 0; i < n; ++i) {
    if (M.count(a[i])) {
      vector<int> ys = M[a[i]];
      for (int j = 0; j < (int)ys.size(); ++j) {
        int k = index_of(xs, ys[j]);
        xs[k] = ys[j];
        link[k] = new node(b[ys[j]], link[k - 1]);
      }
    }
  }
  vector<int> c;
  int l = index_of(xs, INF - 1) - 1;
  for (node *p = link[l]; p; p = p->next) c.push_back(p->value);
  reverse(c.begin(), c.end());
  return c;
}
bool IsPrime(ll n) {
  if (n < 2) return false;
  for (ll i = 2; i * i <= n; i++)
    if (!(n % i)) return false;
  return true;
}
vector<bool> Eratosthenes(int n) {
  vector<int> res;
  vector<bool> Prime(n + 1, true);
  Prime[0] = Prime[1] = false;
  for (int i = 2; i * i <= n; i++) {
    if (Prime[i]) {
      for (int j = 2; i * j <= n; j++) {
        Prime[i * j] = false;
      }
    }
  }
  for (int i = 2; i <= n; i++) {
    if (Prime[i]) {
      res.emplace_back(i);
    }
  }
  return Prime;
}
ll MergeCount(vector<int> &a) {
  ll count = 0;
  int n = a.size();
  if (n > 1) {
    vector<int> b(a.begin(), a.begin() + n / 2);
    vector<int> c(a.begin() + n / 2, a.end());
    count += MergeCount(b);
    count += MergeCount(c);
    for (int i = 0, j = 0, k = 0; i < n; ++i)
      if (k == (int)c.size())
        a[i] = b[j++];
      else if (j == (int)b.size())
        a[i] = c[k++];
      else if (b[j] <= c[k])
        a[i] = b[j++];
      else {
        a[i] = c[k++];
        count += n / 2 - j;
      }
  }
  return count;
}
bool WarshallFloyd(vector<vector<ll>> &c) {
  int V = c.size();
  for (int i = 0; i < V; i++) {
    c[i][i] = 0;
  }
 
  for (int i = 0; i < V; i++) {
    for (int j = 0; j < V; j++) {
      for (int k = 0; k < V; k++) {
        if (c[j][k] > c[j][i] + c[i][k]) {
          c[j][k] = c[j][i] + c[i][k];
        }
      }
    }
  }
 
  for (int i = 0; i < V; i++) {
    if (c[i][i] < 0) {
      return false;
    }
  }
 
  return true;
}
vector<ll> Dijkstra(int i, vector<vector<edge>> graph) {
  int n = graph.size();
  vector<ll> d(n, LONGINF);
  d[i] = 0;
  priority_queue<pair<ll, int>, vector<pair<ll, int>>, greater<pair<ll, int>>>
      q;
  q.push(make_pair(0, i));  
  while (!q.empty()) {
    pair<ll, int> p = q.top();
    q.pop();
    int v = p.second;
    if (d[v] < p.first) {
      continue;
    }
    for (auto x : graph[v]) {
      if (d[x.to] > d[v] + x.cost) {
        d[x.to] = d[v] + x.cost;
        q.push(make_pair(d[x.to], x.to));
      }
    }
  }
  return d;
}
bool BellmanFord(int start, int V, int E, vector<edge> Edge, vector<ll> &d) {
  resize(d, V);
  fill(d.begin(), d.end(), LONGINF);
  d[start] = 0;
  vector<bool> t(V, false);
 
  for (int i = 0; i < V - 1; i++) {
    for (int j = 0; j < E; j++) {
      edge e = Edge[j];
      if (d[e.from] == LONGINF) {
        continue;
      }
      if (d[e.to] > d[e.from] + e.cost) {
        d[e.to] = d[e.from] + e.cost;
      }
    }
  }
 
  for (int i = 0; i < V; i++) {
    for (int j = 0; j < E; j++) {
      edge e = Edge[j];
      if (d[e.from] == LONGINF) {
        continue;
      }
      if (d[e.to] > d[e.from] + e.cost) {
        d[e.to] = d[e.from] + e.cost;
        t[e.to] = true;
        /*
        if (i == V - 1) {
                return false;
        }
        */
      }
      if (t[e.from]) {
        t[e.to] = true;
      }
    }
  }
 
  if (t[V - 1]) {
    return false;
  }
 
  return true;
}
bool TopologicalSort(const vector<vector<edge>> &g, vector<int> &ans) {
  int n = g.size(), k = 0;
  vector<int> ord(n), in(n);
  for (auto &es : g) {
    for (auto &e : es) {
      in[e.to]++;
    }
  }
  queue<int> q;
  for (int i = 0; i < n; ++i) {
    if (in[i] == 0) q.push(i);
  }
  while (!q.empty()) {
    int v = q.front();
    q.pop();
    ord[k++] = v;
    for (auto &e : g[v]) {
      if (--in[e.to] == 0) q.push(e.to);
    }
  }
  ans = ord;
  if (*max_element(in.begin(), in.end()) == 0) {
    return true;
  }
  return false;
}
vector<int> ArticulationNode(const vector<vector<edge>> &g) {
  int n = g.size(), idx;
  vector<int> low(n), ord(n), art;
  function<void(int)> DFS = [&](int v) {
    low[v] = ord[v] = ++idx;
    for (auto &e : g[v]) {
      int w = e.to;
      if (ord[w] == 0) {
        DFS(w);
        low[v] = min(low[v], low[w]);
        if ((ord[v] == 1 && ord[w] != 2) || (ord[v] != 1 && low[w] >= ord[v])) {
          art.push_back(v);
        }
      } else {
        low[v] = min(low[v], ord[w]);
      }
    }
  };
  for (int u = 0; u < n; u++) {
    if (ord[u] == 0) {
      idx = 0;
      DFS(u);
    }
  }
 
  sort(art.begin(), art.end()); 
  art.erase(unique(art.begin(), art.end()),
            art.end()); 
 
  return art;
}
vector<vector<edge>> ToRootTree(const vector<vector<edge>> &g, int r) {
  int n = g.size();
  vector<vector<edge>> G(n);
  vector<int> ord(n, -1);
 
  queue<int> q;
 
  q.push(r);
  int k = 0;
 
  while (q.size()) {
    int u = q.front();
    q.pop();
 
    for (auto &e : g[u]) {
      int v = e.to;
      if (ord[v] == -1) {
        ord[v] = k;
        k++;
        q.push(v);
        G[u].emplace_back(e);
      }
    }
  }
 
  return G;
}
edge TreeDiameter(const vector<vector<edge>> &g) {
 
  int start = 0;  
 
  static const auto bfs = [](const vector<vector<edge>> &g, int s,
                             queue<int> &q, vector<ll> &dist) {
    while (!q.empty()) {
      q.pop();
    }
    q.push(s);
    int n = g.size();
    dist.assign(n, LONGINF);
    dist[s] = 0;
    while (q.size()) {
      int u = q.front();
      q.pop();
      for (auto &e : g[u]) {
        int v = e.to;
        if (dist[v] == LONGINF) {
          dist[v] = dist[u] + e.cost;
          q.push(v);
        }
      }
    }
    return dist;
  };
  vector<ll> dist;
  queue<int> q;
  bfs(g, start, q, dist);
  int n = g.size(), u = -1, v = -1;
  for (int i = 0; i < n; i++)
    if (dist[i] != LONGINF && (u == -1 || dist[i] > dist[u])) u = i;
  bfs(g, u, q, dist);
  for (int i = 0; i < n; i++)
    if (dist[i] != LONGINF && (v == -1 || dist[i] > dist[v])) v = i;
  ll d = dist[v];
  if (u > v) swap(u, v); 
  return edge(u, v, d);
}
void add_edge(vector<vector<edge>> &g, int a, int b, ll cost, ll cap) {
  g[a].emplace_back(a, b, cost, cap);
  g[b].emplace_back(b, a, cost, cap);
}
pair<vector<int>, vector<edge>> bridge(const vector<vector<edge>> &g) {
  const int n = g.size();
  int idx = 0, s = 0, t = 0, k = 0;
  vector<int> ord(n, -1), onS(n), stk(n), roots(n), cmp(n);
  vector<edge> brdg;
  function<void(int, int)> dfs = [&](int v, int u) {
    ord[v] = idx++;
    stk[s++] = v;
    onS[v] = true;
    roots[t++] = v;
    for (auto &e : g[v]) {
      int w = e.to;
      if (ord[w] == -1) {
        dfs(w, v);
      } else if (u != w && onS[w]) {
        while (ord[roots[t - 1]] > ord[w]) {
          --t;
        }
      }
    }
    if (v == roots[t - 1]) {
      brdg.emplace_back(u, v, 0);
      while (1) {
        int w = stk[--s];
        onS[w] = false;
        cmp[w] = k;
        if (v == w) break;
      }
      --t;
      ++k;
    }
  };
  for (int u = 0; u < n; u++) {
    if (ord[u] == -1) {
      dfs(u, n);
      brdg.pop_back();
    }
  }
  return make_pair(cmp, brdg);
}
 
class UnionFind {
 private:
  std::vector<int> parent;
  std::vector<int> height;
  std::vector<int> m_size;
  int forest_num;
 
 public:
  UnionFind(int size_) : parent(size_), height(size_, 0), m_size(size_, 1) {
    forest_num = size_;
    for (int i = 0; i < size_; ++i) parent[i] = i;
  }
  void init(int size_) {
    parent.resize(size_);
    height.resize(size_, 0);
    m_size.resize(size_, 1);
    forest_num = size_;
    for (int i = 0; i < size_; ++i) parent[i] = i;
  }
  int find(int x) {
    if (parent[x] == x) return x;
    return parent[x] = find(parent[x]);
  }
  void unite(int x, int y) {
    x = find(x);
    y = find(y);
    if (x == y) return;
    int t = size(x) + size(y);
    m_size[x] = m_size[y] = t;
    if (height[x] < height[y])
      parent[x] = y;
    else
      parent[y] = x;
    if (height[x] == height[y]) ++height[x];
    forest_num--;
  }
  bool same(int x, int y) { return find(x) == find(y); }
  int size(int x) {
    if (parent[x] == x) return m_size[x];
    return size(parent[x] = find(parent[x]));
  }
  int forest() { return forest_num; }
};
class Dinic {
 private:
  int n, s, t;
  vector<int> level, prog, que;
  vector<vector<ll>> cap, flow;
  vector<vector<int>> g;
  ll inf;
 
 public:
  Dinic(const vector<vector<edge>> &graph)
      : n(graph.size()),
        cap(n, vector<ll>(n)), 
        flow(n, vector<ll>(n)),
        g(n, vector<int>()),
        inf(LONGINF) {
    for (int i = 0; i < n; i++) {
      for (auto &e : graph[i]) {
        int u = e.from, v = e.to;
        ll c = e.capa;
        cap[u][v] += c;
        cap[v][u] += c;
        flow[v][u] += c;
        g[u].push_back(v);
        g[v].push_back(u);
      }
    }
  }
  inline ll residue(int u, int v) { return cap[u][v] - flow[u][v]; }
  ll solve(int s_, int t_) {
    this->t = t_, this->s = s_;
    que.resize(n + 1);
    ll res = 0;
    while (levelize()) {
      prog.assign(n, 0);
      res += augment(s, inf);
    }
    return res;
  }
  bool levelize() {
    int l = 0, r = 0;
    level.assign(n, -1);
    level[s] = 0;
    que[r++] = s;
    while (l != r) {
      int v = que[l++];
      if (v == t) break;
      for (const int &d : g[v])
        if (level[d] == -1 && residue(v, d) != 0) {
          level[d] = level[v] + 1;
          que[r++] = d;
        }
    }
    return level[t] != -1;
  }
  ll augment(int v, ll lim) {
    ll res = 0;
    if (v == t) return lim;
    for (int &i = prog[v]; i < (int)g[v].size(); i++) {
      const int &d = g[v][i];
      if (residue(v, d) == 0 || level[v] >= level[d]) continue;
      const ll aug = augment(d, min(lim, residue(v, d)));
      flow[v][d] += aug;
      flow[d][v] -= aug;
      res += aug;
      lim -= aug;
      if (lim == 0) break;
    }
    return res;
  }
};
class MinimumCostFlow {
 private:
  using Flow = ll;
  using Cost = ll;
  struct Edge {
    int d;
    Flow c, f;
    Cost w;
    int r, is_r;
    Edge(int d_, Flow c_, Flow f_, Cost w_, int r_, bool is_r_)
        : d(d_), c(c_), f(f_), w(w_), r(r_), is_r(is_r_) {}
  };
  int n;
  vector<vector<Edge>> g;
 
 public:
  MinimumCostFlow(int n_) : n(n_), g(vector<vector<Edge>>(n_)) {}
 
  void add_edge(int src, int dst, Flow cap, Cost cost) { 
    int rsrc = g[dst].size();
    int rdst = g[src].size();
    g[src].emplace_back(dst, cap, 0, cost, rsrc, false);
    g[dst].emplace_back(src, cap, cap, -cost, rdst, true);
  }
 
  Cost solve(int s, int t, Flow f) {
    Cost res = 0;
 
    vector<Cost> h(n + 10), dist(n);
    vector<int> prevv(n + 10), preve(n + 10);
 
    using pcv = pair<Cost, int>;
    priority_queue<pcv, vector<pcv>, greater<pcv>> q;
    fill(h.begin(), h.end(), 0);
    while (f > 0) {
      fill(dist.begin(), dist.end(), LONGINF);
      dist[s] = 0;
      q.emplace(0, s);
      while (q.size()) {
        Cost cd;
        int v;
        tie(cd, v) = q.top();
        q.pop();
        if (dist[v] < cd) continue;
        for (int i = 0; i < (int)(g[v].size()); ++i) {
          Edge &e = g[v][i];
          if (residue(e) == 0) continue;
          if (dist[e.d] + h[e.d] > cd + h[v] + e.w) {
            dist[e.d] = dist[v] + e.w + h[v] - h[e.d];
            prevv[e.d] = v;
            preve[e.d] = i;
            q.emplace(dist[e.d], e.d);
          }
        }
      }
 
      if (dist[t] == LONGINF) return -1; 
 
      for (int i = 0; i < n; ++i) h[i] += dist[i];
      Flow d = f;
      for (int v = t; v != s; v = prevv[v]) {
        chmin(d, residue(g[prevv[v]][preve[v]]));
      }
      f -= d;
      res += d * h[t];
      for (int v = t; v != s; v = prevv[v]) {
        Edge &e = g[prevv[v]][preve[v]];
        e.f += d;
        g[v][e.r].f -= d;
      }
    }
    return res;
  }
 
  Flow residue(const Edge &e) { return e.c - e.f; }
 
  void show() {
    for (int i = 0; i < n; ++i) {
      for (int j = 0; j < (int)(g[i].size()); ++j) {
        Edge &e = g[i][j];
        if (e.is_r) continue;
        cout << i << "->" << e.d << "(flow:" << e.f << ")" << endl;
      }
    }
  }
};
class BipartiteMatching {
 private:
  int V;
  vector<int> match;
  vector<bool> used;
  vector<vector<int>> g;
  vector<pair<int, int>> match_pair;
 
  bool dfs(int v) {
    used[v] = true;
    for (int i = 0; i < (int)g[v].size(); i++) {
      int u = g[v][i];
      int w = match[u];
      if (w < 0 || !used[w] && dfs(w)) {
        match[v] = u;
        match[u] = v;
        match_pair.emplace_back(make_pair(u, v));
        return true;
      }
    }
    return false;
  }
 
 public:
  BipartiteMatching(int n) {
    V = n;
    resize(match, n);
    resize(used, n);
    resize(g, n);
  }
 
  void add_edge(int u, int v) {
    g[u].emplace_back(v);
    g[v].emplace_back(u);
  }
 
  int MatchingSolve() {
    int res = 0;
    fill(match.begin(), match.end(), -1);
 
    for (int v = 0; v < V; v++) {
      if (match[v] < 0) {
        fill(used.begin(), used.end(), false);
        if (dfs(v)) {
          res++;
        }
      }
    }
    return res;
  }
 
  vector<pair<int, int>> get_pair() {
    for (auto x : match_pair) {
      cout << x.first << "  " << x.second << endl;
    }
    return match_pair;
  }
};
class Lca {
 private:
  int n;
  int log2_n;
  vector<vector<int>> parent;
  vector<int> depth;
 
  void dfs(const vector<vector<edge>> &g, int v, int p, int d) {
    parent[0][v] = p;
    depth[v] = d;
    for (auto &e : g[v]) {
      if (e.to != p) {
        dfs(g, e.to, v, d + 1);
      }
    }
  }
 
 public:
  Lca(const vector<vector<edge>> &g, int root) {
    n = g.size();
    log2_n = (int)log2(n) + 1;
    resize(parent, log2_n, n);
    resize(depth, n);
 
    dfs(g, root, -1, 0);
 
    for (int k = 0; k + 1 < log2_n; k++) {
      for (int v = 0; v < (int)g.size(); v++) {
        if (parent[k][v] < 0) {
          parent[k + 1][v] = -1;
        } else {
          parent[k + 1][v] = parent[k][parent[k][v]];
        }
      }
    }
  }
 
  int get_lca(int u, int v) {
    if (depth[u] > depth[v]) {
      swap(u, v);
    }  
 
    for (int k = 0; k < log2_n; k++) {
      if ((depth[v] - depth[u]) >> k & 1) {
        v = parent[k][v];
      }
    }
    if (u == v) {
      return u;
    }
 
    for (int k = log2_n - 1; k >= 0; k--) {
      if (parent[k][u] != parent[k][v]) {
        u = parent[k][u];
        v = parent[k][v];
      }
    }
    return parent[0][u];
  }
 
  int get_depth(int v) { return depth[v]; }
};
class DAG {
 private:
  int n;
  vector<vector<edge>> g;
  vector<int> visited;
  vector<int> dp;
  vector<int> topological;
 
  int dfs(int s) {
    if ((int)g[s].size() == 0) {
      return 1;
    }
    if (dp[s] > 0) {
      return dp[s];
    }
 
    int mx = 1;
    for (auto j : g[s]) {
      if (visited[j.to] == 0) {
        visited[j.to] = 1;
        int l = 0;
        l = dfs(j.to);
        chmax(mx, l);
      } else {
        chmax(mx, dp[j.to]);
      }
    }
    return dp[s] = mx + 1;
  }
 
 public:
  DAG(const vector<vector<edge>> &f) {
    g = f;
    n = f.size();
    resize(visited, n + 1);
    fill(visited.begin(), visited.end(), 0);
    resize(dp, n + 1);
    fill(dp.begin(), dp.end(), -1);
    resize(topological, n);
  }
  DAG(int x) {
    n = x;
    resize(g, n);
    resize(visited, n + 1);
    fill(visited.begin(), visited.end(), 0);
    resize(dp, n + 1);
    fill(dp.begin(), dp.end(), -1);
  }
 
  void add_edge(int a, int b) { g[a].emplace_back(a, b); }
  void add_edge(int a, int b, ll c) { g[a].emplace_back(a, b, c); }
  void add_edge(int a, int b, ll c, ll d) { g[a].emplace_back(a, b, c, d); }
 
  int longest_path() {
    int mx = -1;
    for (int i = 0; i < n; i++) {
      int h = 0;
      if (visited[i] == 0) {
        h = dfs(i);
        chmax(mx, h);
      }
    }
    return mx - 1;
  }
 
  bool TopologicalSort() {
    int k = 0;
    vector<int> ord(n), in(n);
    for (auto &es : g) {
      for (auto &e : es) {
        in[e.to]++;
      }
    }
    queue<int> q;
    for (int i = 0; i < n; ++i) {
      if (in[i] == 0) q.push(i);
    }
    while (!q.empty()) {
      int v = q.front();
      q.pop();
      ord[k++] = v;
      for (auto &e : g[v]) {
        if (--in[e.to] == 0) {
          q.push(e.to);
        }
      }
    }
    topological = ord;
    if (*max_element(in.begin(), in.end()) == 0) {
      return true;
    }
    return false;
  }
};
class RangeMinimumUpdateQuerySegmentTree {
 private:
  int n;
  ll inf = (1LL << 31) - 1;  // 2^31-1
  vector<ll> dat, lazy;
 
  void eval(int len, int k) {
    if (lazy[k] == inf) return;
    if (k * 2 + 1 < n * 2 - 1) {
      lazy[2 * k + 1] = lazy[k];
      lazy[2 * k + 2] = lazy[k];
    }
    dat[k] = lazy[k];
    lazy[k] = inf;
  }
 
 public:
  RangeMinimumUpdateQuerySegmentTree() {}
  RangeMinimumUpdateQuerySegmentTree(int n_) {
    n = 1;
    while (n < n_) n *= 2;
    dat.assign(n * 2, inf);
    lazy.assign(n * 2, inf);
  }
 
  // [a,b)
  ll update(int a, int b, ll x, int k, int l, int r) {
    eval(r - l, k);
    if (b <= l || r <= a) return dat[k];
    if (a <= l && r <= b) {
      lazy[k] = x;
      return lazy[k];
    }
    return dat[k] = min(update(a, b, x, 2 * k + 1, l, (l + r) / 2),
                        update(a, b, x, 2 * k + 2, (l + r) / 2, r));
  }
  ll update(int a, int b, ll x) { return update(a, b, x, 0, 0, n); }
 
  // [a, b)
  ll query(int a, int b, int k, int l, int r) {
    eval(r - l, k);
    if (b <= l || r <= a) return inf;
    if (a <= l && r <= b) return dat[k];
    ll vl = query(a, b, 2 * k + 1, l, (l + r) / 2);
    ll vr = query(a, b, 2 * k + 2, (l + r) / 2, r);
    return min(vl, vr);
  }
  ll query(int a, int b) { return query(a, b, 0, 0, n); }
};
class RangeSumQuerySegmentTree {
 private:
  struct Node {
    Node *left, *right;
    ll v;
 
    Node() : left(nullptr), right(nullptr), v(0) {}
  };
  Node *root;
  ll n, n0;
  ll query(ll a, ll b, Node *n, ll l, ll r) {
    if (a <= l && r <= b) {
      return n->v;
    }
    if (r <= a || b <= l) {
      return 0;
    }
 
    ll lv = n->left ? query(a, b, n->left, l, (l + r) >> 1) : 0;
    ll rv = n->right ? query(a, b, n->right, (l + r) >> 1, r) : 0;
    return (lv + rv) % MOD;
  }
 
 public:
  RangeSumQuerySegmentTree(ll n) : n(n) {
    n0 = 1;
    while (n0 < n) n0 <<= 1;
    root = new Node();
  }
  ~RangeSumQuerySegmentTree() {
    delete root;
    root = nullptr;
  }
 
  void update(ll k, ll x) {
    Node *n = root;
    ll l = 0, r = n0;
    n->v = (n->v + x) % MOD;
    while (r - l > 1) {
      ll m = (l + r) >> 1;
      if (k < m) {
        if (!n->left) n->left = new Node();
        n = n->left;
 
        r = m;
      } else {
        if (!n->right) n->right = new Node();
        n = n->right;
 
        l = m;
      }
      n->v = (n->v + x) % MOD;
    }
  }
 
  ll query(ll a, ll b) { return query(a, b, root, 0, n0); }
 
  ll lquery(ll b) { return query(0, b, root, 0, n0); }
 
  ll rquery(ll a) { return query(a, n0, root, 0, n0); }
};
class KDimensionalTree {
 public:
  struct Node {
    int location;
    int p, l, r;
    Node() {}
  };
  struct Point {
    int id, x, y;
    Point() {}
    Point(int i, int a, int b) {
      id = i;
      x = a;
      y = b;
    }
    bool operator<(const Point &p) const { return id < p.id; }
    void print() { cout << id << endl; }
  };
  static const ll NIL = -1;
  static bool lessX(const Point &p1, const Point &p2) { return p1.x < p2.x; }
  static bool lessY(const Point &p1, const Point &p2) { return p1.y < p2.y; }
 
  int N;
  vector<Point> P;
  vector<Node> T;
  int np;
 
  KDimensionalTree() {}
  KDimensionalTree(int N) { init(N); }
 
  void init(int n) {
    N = n;
    P.clear();
    T.clear();
    resize(P, N);
    resize(T, N);
    np = 0;
  }
 
  int makeKDTree(int l, int r, int depth) {
    if (l >= r) {
      return NIL;
    }
    int mid = (l + r) / 2;
    int t = np++;
    if (depth & 1) {
      sort(P.begin() + l, P.begin() + r, lessY);
    } else {
      sort(P.begin() + l, P.begin() + r, lessX);
    }
    T[t].location = mid;
    T[t].l = makeKDTree(l, mid, depth + 1);
    T[t].r = makeKDTree(mid + 1, r, depth + 1);
    return t;
  }
  void find(int v, int sx, int tx, int sy, int ty, int depth,
            vector<Point> &ans) {
    int x = P[T[v].location].x;
    int y = P[T[v].location].y;
    if (sx <= x && x <= tx && sy <= y && y <= ty) {
      ans.push_back(P[T[v].location]);
    }
    if (depth % 2 == 0) {
      if (T[v].l != NIL) {
        if (sx <= x) find(T[v].l, sx, tx, sy, ty, depth + 1, ans);
      }
      if (T[v].r != NIL) {
        if (x <= tx) find(T[v].r, sx, tx, sy, ty, depth + 1, ans);
      }
    } else {
      if (T[v].l != NIL) {
        if (sy <= y) find(T[v].l, sx, tx, sy, ty, depth + 1, ans);
      }
      if (T[v].r != NIL) {
        if (y <= ty) find(T[v].r, sx, tx, sy, ty, depth + 1, ans);
      }
    }
  }
  void add_point(int i, int x, int y) {
    P[i].id = i;
    P[i].x = x;
    P[i].y = y;
    T[i].l = T[i].r = T[i].p = NIL;
  }
};
class RangeAddQuerySegmentTree {
 private:
  int n;
  vector<ll> data;
 
 public:
  RangeAddQuerySegmentTree() {}
  RangeAddQuerySegmentTree(int N) {
    n = N;
    resize(data, n + 1);
    fill(data.begin(), data.end(), 0);
  }
 
  void add(int i, ll x) {
    while (i) {
      data[i] += x;
      i -= (i & -i);
    }
  }
  void add(int i, int j, ll x) {
    add(j, x);
    add(i - 1, -x);
  }
 
  ll get(int i) {
    ll res = 0;
    while (i <= n) {
      res += data[i];
      i += (i & -i);
    }
    return res;
  }
};
class RangeSumAddQuerySegmentTree {
 private:
  vector<ll> bit0, bit1;
  int n;
 
  ll sum(const vector<ll> &b, int i) {
    ll s = 0;
    while (i > 0) {
      s += b[i];
      i -= (i & -i);
    }
    return s;
  }
 
  void add(vector<ll> &b, int i, ll v) {
    while (i <= n) {
      b[i] += v;
      i += (i & -i);
    }
  }
 
 public:
  RangeSumAddQuerySegmentTree() {}
  RangeSumAddQuerySegmentTree(int N) {
    n = N;
    resize(bit0, n + 1);
    resize(bit1, n + 1);
    fill(bit0.begin(), bit0.end(), 0);
    fill(bit1.begin(), bit1.end(), 0);
  }
 
  void update(int l, int r, ll x) {
    add(bit0, l, -x * (l - 1));
    add(bit1, l, x);
    add(bit0, r + 1, x * r);
    add(bit1, r + 1, -x);
  }
 
  ll query(int l, int r) {
    ll res = 0;
    res += sum(bit0, r) + sum(bit1, r) * r;
    res -= sum(bit0, l - 1) + sum(bit1, l - 1) * (l - 1);
    return res;
  }
};


const int MAX = 1010000;

long long fac[MAX], finv[MAX], inv[MAX];

// テーブルを作る前処理
void COMinit() {
    fac[0] = fac[1] = 1;
    finv[0] = finv[1] = 1;
    inv[1] = 1;
    for (int i = 2; i < MAX; i++){
        fac[i] = fac[i - 1] * i % MOD;
        inv[i] = MOD - inv[MOD%i] * (MOD / i) % MOD;
        finv[i] = finv[i - 1] * inv[i] % MOD;
    }
}

// 二項係数計算
long long COM(int n, int k){
    if (n < k) return 0;
    if (n < 0 || k < 0) return 0;
    return fac[n] * (finv[k] * finv[n - k] % MOD) % MOD;
}

int main() {
  int x,y;
  cin >> x >> y;
  COMinit();

  int k = 0, l = 0;

  while(x>0&&y>0){
    if(x>y){
      k++;
      x -= 2;
      y -= 1;
    }
    else{
      l++;
      x -= 1;
      y -= 2;
    }
  }
  if(x==0&&y==0){
    if(k==0||l==0){
      cout << 1 << endl;
    }
    else{
      cout << COM(k + l, k) << endl;
    }
  }
  else{
    cout << 0 << endl;
  }
  //cout << x << " " << y << endl;
  //cout << k << " " << l << endl;

  ;
  return 0;
}

Measured memory_bytes: 23936