#include <bits/stdc++.h>
#ifndef ATCODER_LAZYSEGTREE_HPP
#define ATCODER_LAZYSEGTREE_HPP 1

#ifndef ATCODER_INTERNAL_BITOP_HPP
#define ATCODER_INTERNAL_BITOP_HPP 1

#ifdef _MSC_VER
#include <intrin.h>
#endif

namespace atcoder {

namespace internal {

// @param n `0 <= n`
// @return minimum non-negative `x` s.t. `n <= 2**x`
int ceil_pow2(int n) {
    int x = 0;
    while ((1U << x) < (unsigned int)(n)) x++;
    return x;
}

// @param n `1 <= n`
// @return minimum non-negative `x` s.t. `(n & (1 << x)) != 0`
constexpr int bsf_constexpr(unsigned int n) {
    int x = 0;
    while (!(n & (1 << x))) x++;
    return x;
}

// @param n `1 <= n`
// @return minimum non-negative `x` s.t. `(n & (1 << x)) != 0`
int bsf(unsigned int n) {
#ifdef _MSC_VER
    unsigned long index;
    _BitScanForward(&index, n);
    return index;
#else
    return __builtin_ctz(n);
#endif
}

}  // namespace internal

}  // namespace atcoder

#endif  // ATCODER_INTERNAL_BITOP_HPP

namespace atcoder {

template <class S,
          S (*op)(S, S),
          S (*e)(),
          class F,
          S (*mapping)(F, S),
          F (*composition)(F, F),
          F (*id)()>
struct lazy_segtree {
  public:
    lazy_segtree() : lazy_segtree(0) {}
    explicit lazy_segtree(int n) : lazy_segtree(std::vector<S>(n, e())) {}
    explicit lazy_segtree(const std::vector<S>& v) : _n(int(v.size())) {
        log = internal::ceil_pow2(_n);
        size = 1 << log;
        d = std::vector<S>(2 * size, e());
        lz = std::vector<F>(size, id());
        for (int i = 0; i < _n; i++) d[size + i] = v[i];
        for (int i = size - 1; i >= 1; i--) {
            update(i);
        }
    }

    void set(int p, S x) {
        assert(0 <= p && p < _n);
        p += size;
        for (int i = log; i >= 1; i--) push(p >> i);
        d[p] = x;
        for (int i = 1; i <= log; i++) update(p >> i);
    }

    S get(int p) {
        assert(0 <= p && p < _n);
        p += size;
        for (int i = log; i >= 1; i--) push(p >> i);
        return d[p];
    }

    S prod(int l, int r) {
        assert(0 <= l && l <= r && r <= _n);
        if (l == r) return e();

        l += size;
        r += size;

        for (int i = log; i >= 1; i--) {
            if (((l >> i) << i) != l) push(l >> i);
            if (((r >> i) << i) != r) push((r - 1) >> i);
        }

        S sml = e(), smr = e();
        while (l < r) {
            if (l & 1) sml = op(sml, d[l++]);
            if (r & 1) smr = op(d[--r], smr);
            l >>= 1;
            r >>= 1;
        }

        return op(sml, smr);
    }

    S all_prod() { return d[1]; }

    void apply(int p, F f) {
        assert(0 <= p && p < _n);
        p += size;
        for (int i = log; i >= 1; i--) push(p >> i);
        d[p] = mapping(f, d[p]);
        for (int i = 1; i <= log; i++) update(p >> i);
    }
    void apply(int l, int r, F f) {
        assert(0 <= l && l <= r && r <= _n);
        if (l == r) return;

        l += size;
        r += size;

        for (int i = log; i >= 1; i--) {
            if (((l >> i) << i) != l) push(l >> i);
            if (((r >> i) << i) != r) push((r - 1) >> i);
        }

        {
            int l2 = l, r2 = r;
            while (l < r) {
                if (l & 1) all_apply(l++, f);
                if (r & 1) all_apply(--r, f);
                l >>= 1;
                r >>= 1;
            }
            l = l2;
            r = r2;
        }

        for (int i = 1; i <= log; i++) {
            if (((l >> i) << i) != l) update(l >> i);
            if (((r >> i) << i) != r) update((r - 1) >> i);
        }
    }

    template <bool (*g)(S)> int max_right(int l) {
        return max_right(l, [](S x) { return g(x); });
    }
    template <class G> int max_right(int l, G g) {
        assert(0 <= l && l <= _n);
        assert(g(e()));
        if (l == _n) return _n;
        l += size;
        for (int i = log; i >= 1; i--) push(l >> i);
        S sm = e();
        do {
            while (l % 2 == 0) l >>= 1;
            if (!g(op(sm, d[l]))) {
                while (l < size) {
                    push(l);
                    l = (2 * l);
                    if (g(op(sm, d[l]))) {
                        sm = op(sm, d[l]);
                        l++;
                    }
                }
                return l - size;
            }
            sm = op(sm, d[l]);
            l++;
        } while ((l & -l) != l);
        return _n;
    }

    template <bool (*g)(S)> int min_left(int r) {
        return min_left(r, [](S x) { return g(x); });
    }
    template <class G> int min_left(int r, G g) {
        assert(0 <= r && r <= _n);
        assert(g(e()));
        if (r == 0) return 0;
        r += size;
        for (int i = log; i >= 1; i--) push((r - 1) >> i);
        S sm = e();
        do {
            r--;
            while (r > 1 && (r % 2)) r >>= 1;
            if (!g(op(d[r], sm))) {
                while (r < size) {
                    push(r);
                    r = (2 * r + 1);
                    if (g(op(d[r], sm))) {
                        sm = op(d[r], sm);
                        r--;
                    }
                }
                return r + 1 - size;
            }
            sm = op(d[r], sm);
        } while ((r & -r) != r);
        return 0;
    }

  private:
    int _n, size, log;
    std::vector<S> d;
    std::vector<F> lz;

    void update(int k) { d[k] = op(d[2 * k], d[2 * k + 1]); }
    void all_apply(int k, F f) {
        d[k] = mapping(f, d[k]);
        if (k < size) lz[k] = composition(f, lz[k]);
    }
    void push(int k) {
        all_apply(2 * k, lz[k]);
        all_apply(2 * k + 1, lz[k]);
        lz[k] = id();
    }
};

}  // namespace atcoder

#endif  // ATCODER_LAZYSEGTREE_HPP

namespace update_min {
using S = int64_t;
S op(S a, S b) { return std::min(a, b); }
S e() { return std::numeric_limits<S>::max(); }
using F = std::optional<int64_t>;
S mapping(F f, S x) { return f ? *f : x; }
F composition(F f, F g) { return f ? f : g; }
F id() { return std::nullopt; }
using segtree = atcoder::lazy_segtree<S, op, e, F, mapping, composition, id>;
}  // namespace update_min

namespace add_min {
using S = int64_t;
S op(S a, S b) { return std::min(a, b); }
S e() { return std::numeric_limits<S>::max(); }
using F = int64_t;
S mapping(F f, S x) { return f + x; }
F composition(F f, F g) { return f + g; }
F id() { return 0; }
using segtree = atcoder::lazy_segtree<S, op, e, F, mapping, composition, id>;
}  // namespace add_min

namespace update_sum {
struct S {
  int64_t val;
  int64_t len;
};
S op(S a, S b) { return {a.val + b.val, a.len + b.len}; }
S e() { return {0, 0}; }
using F = std::optional<int64_t>;
S mapping(F f, S x) {
  if (f) x.val = *f * x.len;
  return x;
}
F composition(F f, F g) { return f ? f : g; }
F id() { return std::nullopt; }
class segtree
    : public atcoder::lazy_segtree<S, op, e, F, mapping, composition, id> {
 public:
  segtree() : lazy_segtree() {}
  explicit segtree(int n) : lazy_segtree(std::vector<S>(n, {0, 1})) {}
  explicit segtree(const std::vector<int64_t>& v) : lazy_segtree(itos(v)) {}

 private:
  static std::vector<S> itos(const std::vector<int64_t>& v) {
    std::vector<S> w(v.size());
    for (size_t i = 0; i < v.size(); ++i) {
      w[i] = {v[i], 1};
    }
    return w;
  }
};
}  // namespace update_sum

namespace add_sum {
struct S {
  int64_t val;
  int64_t len;
};
S op(S a, S b) { return {a.val + b.val, a.len + b.len}; }
S e() { return {0, 0}; }
using F = int64_t;
S mapping(F f, S x) { return {x.val + f * x.len, x.len}; }
F composition(F f, F g) { return f + g; }
F id() { return 0; }
class segtree
    : public atcoder::lazy_segtree<S, op, e, F, mapping, composition, id> {
 public:
  segtree() : lazy_segtree() {}
  explicit segtree(int n) : lazy_segtree(std::vector<S>(n, {0, 1})) {}
  explicit segtree(const std::vector<int64_t>& v) : lazy_segtree(itos(v)) {}

 private:
  static std::vector<S> itos(const std::vector<int64_t>& v) {
    std::vector<S> w(v.size());
    for (size_t i = 0; i < v.size(); ++i) {
      w[i] = {v[i], 1};
    }
    return w;
  }
};
}  // namespace add_sum

template <typename T>
class SegmentTree {
 public:
  using Operation = std::function<T(T, T)>;
  SegmentTree(int size, Operation operation, T identity = T())
      : operation_(operation), identity_(identity) {
    int two = 1;
    while (two < size) {
      two <<= 1;
    }
    v_.resize(two * 2 - 1, identity_);
  }
  void Set(int i, T v) {
    int index = Leaf(i);
    while (true) {
      v_[index] = v;
      if (index == 0) break;
      v = operation_(v, v_[index + (IsRight(index) ? -1 : 1)]);
      index = Parent(index);
    }
  }
  T Get(int i) const { return Aggregate(i, i + 1); }
  T Aggregate(int begin, int end) const {
    int l = Leaf(begin), r = Leaf(end);
    T v = identity_;
    while (l < r) {
      if (IsRight(l)) {
        v = operation_(v, v_[l]);
        ++l;
      }
      l = Parent(l);
      if (IsRight(r)) {
        v = operation_(v, v_[r - 1]);
      }
      r = Parent(r);
    }
    return v;
  }

 private:
  int Leaf(int i) const { return i + (v_.size() >> 1); }
  bool IsRight(int i) const { return !(i & 1); }
  int Parent(int i) const { return (i - 1) >> 1; }
  const Operation operation_;
  const T identity_;
  std::vector<T> v_;
};

template <typename T>
class AddSegmentTree : public SegmentTree<T> {
 public:
  AddSegmentTree(int n) : SegmentTree<T>(n, [](T a, T b) { return a + b; }) {}
};

#ifndef GRAPH_H_
#define GRAPH_H_

template <typename T>
class Graph {
 public:
  struct Edge {
    int from, to;
    T weight;
  };

  Graph(int n) : edges_(n) {}
  void AddEdge(int from, int to, T weight = T()) {
    edges_[from].push_back({from, to, weight});
  }
  const std::vector<Edge> &Edges(int from) const { return edges_[from]; }
  std::vector<Edge> &MutableEdges(int from) { return edges_[from]; }
  int NumVertices() const { return edges_.size(); }
  bool IsTree() const {
    std::vector<bool> visited(NumVertices());
    auto rec = [&](auto rec, int node, int parent) -> bool {
      if (visited[node]) return false;
      visited[node] = true;
      for (const Edge &e : Edges(node)) {
        if (e.to != parent && !rec(rec, e.to, node)) {
          return false;
        }
      }
      return true;
    };
    return rec(rec, 0, -1);
  }

 private:
  std::vector<std::vector<Edge>> edges_;
};

#endif

template <typename T>
class HeavyLightDecomposition {
 public:
  HeavyLightDecomposition(const Graph<T>& g, int root = 0) : g_(g) {
#ifdef DEBUG
    assert(g.IsTree());
#endif
    int n = g.NumVertices();
    attr_.resize(n);
    Dfs1(root, -1, 0);
    int index = 0;
    Dfs2(root, -1, root, index);
  }
  std::vector<std::pair<int32_t, int32_t>> Query(
      int u, int v, bool include_lca = false) const {
    std::vector<std::pair<int32_t, int32_t>> ret;
    while (Begin(u) != Begin(v)) {
      if (Depth(Begin(u)) < Depth(Begin(v))) std::swap(u, v);
      ret.emplace_back(Index(Begin(u)), Index(u) + 1);
      u = Parent(Begin(u));
    }
    u = Index(u), v = Index(v);
    if (u > v) std::swap(u, v);
    ret.emplace_back(u + (include_lca ? 0 : 1), v + 1);
    return ret;
  }
  int LCA(int u, int v) const {
    while (Begin(u) != Begin(v)) {
      if (Depth(Begin(u)) < Depth(Begin(v))) std::swap(u, v);
      u = Parent(Begin(u));
    }
    return Depth(u) < Depth(v) ? u : v;
  }
  int32_t Index(int node) const { return attr_[node].index; }
  int32_t Parent(int node) const { return attr_[node].parent; }

 private:
  void Dfs1(int node, int parent, int depth) {
    Attr& a = attr_[node];
    a.depth = depth;
    a.parent = parent;
    a.size = 1;
    a.heavy = -1;
    for (const auto& e : g_.Edges(node)) {
      if (e.to == parent) continue;
      Dfs1(e.to, node, depth + 1);
      a.size += Size(e.to);
      if (a.heavy == -1 || Size(a.heavy) < Size(e.to)) {
        a.heavy = e.to;
      }
    }
  }
  void Dfs2(int node, int parent, int begin, int& index) {
    Attr& a = attr_[node];
    a.index = index++;
    a.begin = begin;
    if (a.heavy == -1) return;
    Dfs2(a.heavy, node, begin, index);
    for (const auto& e : g_.Edges(node)) {
      if (e.to == parent || e.to == a.heavy) continue;
      Dfs2(e.to, node, e.to, index);
    }
  }
  int32_t Begin(int node) const { return attr_[node].begin; }
  int32_t Depth(int node) const { return attr_[node].depth; }
  int32_t Heavy(int node) const { return attr_[node].heavy; }
  int32_t Size(int node) const { return attr_[node].size; }

  const Graph<T>& g_;
  struct Attr {
    int32_t begin;
    int32_t depth;
    int32_t heavy;
    int32_t index;
    int32_t parent;
    int32_t size;
  };
  std::vector<Attr> attr_;
};
#ifndef MODINT_H_
#define MODINT_H_

namespace {
using i32 = int32_t;
using i64 = int64_t;
}  // namespace

#define BIN_OPS(F) F(+) F(-) F(*) F(/)
#define CMP_OPS(F) F(!=) F(<) F(<=) F(==) F(>) F(>=)

template <i32 Mod = 1000000007>
class ModInt {
 public:
  ModInt() : n_(0) {}
  ModInt(i64 n) : n_(n % Mod) {
    if (n_ < 0) {
      // In C++, (-n)%m == -(n%m).
      n_ += Mod;
    }
  }
  ModInt& operator+=(const ModInt& m) {
    n_ += m.n_;
    if (n_ >= Mod) {
      n_ -= Mod;
    }
    return *this;
  }
  ModInt& operator++() { return (*this) += 1; }
  ModInt& operator-=(const ModInt& m) {
    n_ -= m.n_;
    if (n_ < 0) {
      n_ += Mod;
    }
    return *this;
  }
  ModInt& operator--() { return (*this) -= 1; }
  ModInt& operator*=(const ModInt& m) {
    n_ = i64(n_) * m.n_ % Mod;
    return *this;
  }
  ModInt& operator/=(const ModInt& m) {
    *this *= m.Inv();
    return *this;
  }
#define DEFINE(op) \
  ModInt operator op(const ModInt& m) const { return ModInt(*this) op## = m; }
  BIN_OPS(DEFINE)
#undef DEFINE
#define DEFINE(op) \
  bool operator op(const ModInt& m) const { return n_ op m.n_; }
  CMP_OPS(DEFINE)
#undef DEFINE
  ModInt operator-() const { return ModInt(-n_); }
  ModInt Pow(i64 n) const {
    if (n < 0) {
      return Inv().Pow(-n);
    }
    // a * b ^ n = answer.
    ModInt a = 1, b = *this;
    while (n != 0) {
      if (n & 1) {
        a *= b;
      }
      n /= 2;
      b *= b;
    }
    return a;
  }
  ModInt Inv() const {
#if DEBUG
    assert(n_ != 0);
#endif
    if (n_ > kMaxCacheSize) {
      // Compute the inverse based on Fermat's little theorem. Note that this
      // only works when n_ and Mod are relatively prime. The theorem says that
      // n_^(Mod-1) = 1 (mod Mod). So we can compute n_^(Mod-2).
      return Pow(Mod - 2);
    }
    for (i64 i = inv_.size(); i <= n_; ++i) {
      inv_.push_back(i <= 1 ? i : (Mod / i * -inv_[Mod % i]));
    }
    return inv_[n_];
  }
  i64 value() const { return n_; }

  static ModInt Fact(i64 n) {
#if DEBUG
    assert(0 <= n && n <= kMaxCacheSize);
#endif
    for (i64 i = fact_.size(); i <= n; ++i) {
      fact_.push_back(i == 0 ? 1 : fact_.back() * i);
    }
    return fact_[n];
  }
  static ModInt InvFact(i64 n) {
#if DEBUG
    assert(0 <= n && n <= kMaxCacheSize);
#endif
    for (i64 i = inv_fact_.size(); i <= n; ++i) {
      inv_fact_.push_back(i == 0 ? 1 : inv_fact_.back() / i);
    }
    return inv_fact_[n];
  }
  static ModInt Comb(i64 n, i64 k) {
    if (!Valid(n, k)) return 0;
    return Perm(n, k) * InvFact(k);
  }
  static ModInt CombSlow(i64 n, i64 k) {
    if (!Valid(n, k)) return 0;
    return PermSlow(n, k) * InvFact(k);
  }
  static ModInt Perm(i64 n, i64 k) {
    if (!Valid(n, k)) return 0;
#if DEBUG
    assert(n <= kMaxCacheSize &&
           "n is too large. If k is small, consider using PermSlow.");
#endif
    return Fact(n) * InvFact(n - k);
  }
  static ModInt PermSlow(i64 n, i64 k) {
    if (!Valid(n, k)) return 0;
    ModInt p = 1;
    for (i64 i = 0; i < k; ++i) {
      p *= (n - i);
    }
    return p;
  }

 private:
  static bool Valid(i64 n, i64 k) { return 0 <= n && 0 <= k && k <= n; }

  i32 n_;
  inline static std::vector<ModInt> fact_;
  inline static std::vector<ModInt> inv_fact_;
  inline static std::vector<ModInt> inv_;
  static const i64 kMaxCacheSize = 10000000;
};

#define DEFINE(op)                                            \
  template <i32 Mod, typename T>                              \
  ModInt<Mod> operator op(const T& t, const ModInt<Mod>& m) { \
    return ModInt<Mod>(t) op m;                               \
  }
BIN_OPS(DEFINE)
CMP_OPS(DEFINE)
#undef DEFINE

template <i32 Mod>
std::ostream& operator<<(std::ostream& out, const ModInt<Mod>& m) {
  out << m.value();
  return out;
}

#endif
#include <boost/hana/functional/fix.hpp>

template <typename T, typename = void>
struct is_dereferenceable : std::false_type {};
template <typename T>
struct is_dereferenceable<T, std::void_t<decltype(*std::declval<T>())>>
    : std::true_type {};

template <typename T, typename = void>
struct is_iterable : std::false_type {};
template <typename T>
struct is_iterable<T, std::void_t<decltype(std::begin(std::declval<T>())),
                                  decltype(std::end(std::declval<T>()))>>
    : std::true_type {};

template <typename T, typename = void>
struct is_applicable : std::false_type {};
template <typename T>
struct is_applicable<T, std::void_t<decltype(std::tuple_size<T>::value)>>
    : std::true_type {};

template <typename T, typename... Ts>
void debug(const T& value, const Ts&... args);
template <typename T>
void debug(const T& v) {
  if constexpr (is_dereferenceable<T>::value) {
    std::cerr << "{";
    if (v) {
      debug(*v);
    } else {
      std::cerr << "nil";
    }
    std::cerr << "}";
  } else if constexpr (is_iterable<T>::value &&
                       !std::is_same<T, std::string>::value) {
    std::cerr << "{";
    for (auto it = std::begin(v); it != std::end(v); ++it) {
      if (it != std::begin(v)) std::cerr << ", ";
      debug(*it);
    }
    std::cerr << "}";
  } else if constexpr (is_applicable<T>::value) {
    std::cerr << "{";
    std::apply([](const auto&... args) { debug(args...); }, v);
    std::cerr << "}";
  } else {
    std::cerr << v;
  }
}
template <typename T, typename... Ts>
void debug(const T& value, const Ts&... args) {
  debug(value);
  std::cerr << ", ";
  debug(args...);
}
#if DEBUG
#define dbg(...)                        \
  do {                                  \
    cerr << #__VA_ARGS__ << ": ";       \
    debug(__VA_ARGS__);                 \
    cerr << " (L" << __LINE__ << ")\n"; \
  } while (0)
#else
#define dbg(...)
#endif

void read_from_cin() {}
template <typename T, typename... Ts>
void read_from_cin(T& value, Ts&... args) {
  std::cin >> value;
  read_from_cin(args...);
}
#define rd(type, ...) \
  type __VA_ARGS__;   \
  read_from_cin(__VA_ARGS__);
#define ints(...) rd(int, __VA_ARGS__);
#define strings(...) rd(string, __VA_ARGS__);

// Strings used for yes/no questions. Defined as variables so that it can be
// adjusted for each contest site.
const char *yes_str = "Yes", *no_str = "No";

template <typename T>
void write_to_cout(const T& value) {
  if constexpr (std::is_same<T, bool>::value) {
    std::cout << (value ? yes_str : no_str);
  } else if constexpr (is_iterable<T>::value &&
                       !std::is_same<T, std::string>::value) {
    for (auto it = std::begin(value); it != std::end(value); ++it) {
      if (it != std::begin(value)) std::cout << " ";
      std::cout << *it;
    }
  } else {
    std::cout << value;
  }
}
template <typename T, typename... Ts>
void write_to_cout(const T& value, const Ts&... args) {
  write_to_cout(value);
  std::cout << ' ';
  write_to_cout(args...);
}
#define wt(...)                 \
  do {                          \
    write_to_cout(__VA_ARGS__); \
    cout << '\n';               \
  } while (0)

#define all(x) (x).begin(), (x).end()
#define eb(...) emplace_back(__VA_ARGS__)
#define pb(...) push_back(__VA_ARGS__)

#define dispatch(_1, _2, _3, name, ...) name

#define as_i64(x)                                                          \
  (                                                                        \
      [] {                                                                 \
        static_assert(                                                     \
            std::is_integral<                                              \
                typename std::remove_reference<decltype(x)>::type>::value, \
            "rep macro supports std integral types only");                 \
      },                                                                   \
      static_cast<int64_t>(x))

#define rep3(i, a, b) for (int64_t i = as_i64(a); i < as_i64(b); ++i)
#define rep2(i, n) rep3(i, 0, n)
#define rep1(n) rep2(_loop_variable_, n)
#define rep(...) dispatch(__VA_ARGS__, rep3, rep2, rep1)(__VA_ARGS__)

#define rrep3(i, a, b) for (int64_t i = as_i64(b) - 1; i >= as_i64(a); --i)
#define rrep2(i, n) rrep3(i, 0, n)
#define rrep1(n) rrep2(_loop_variable_, n)
#define rrep(...) dispatch(__VA_ARGS__, rrep3, rrep2, rrep1)(__VA_ARGS__)

#define each3(k, v, c) for (auto&& [k, v] : c)
#define each2(e, c) for (auto&& e : c)
#define each(...) dispatch(__VA_ARGS__, each3, each2)(__VA_ARGS__)

template <typename T>
std::istream& operator>>(std::istream& is, std::vector<T>& v) {
  for (T& vi : v) is >> vi;
  return is;
}

template <typename T, typename U>
std::istream& operator>>(std::istream& is, std::pair<T, U>& p) {
  is >> p.first >> p.second;
  return is;
}

template <typename T, typename U>
bool chmax(T& a, U b) {
  if (a < b) {
    a = b;
    return true;
  }
  return false;
}

template <typename T, typename U>
bool chmin(T& a, U b) {
  if (a > b) {
    a = b;
    return true;
  }
  return false;
}

template <typename T, typename U>
auto max(T a, U b) {
  return a > b ? a : b;
}

template <typename T, typename U>
auto min(T a, U b) {
  return a < b ? a : b;
}

template <typename T>
auto max(const T& v) {
  return *std::max_element(v.begin(), v.end());
}

template <typename T>
auto min(const T& v) {
  return *std::min_element(v.begin(), v.end());
}

template <typename T>
int64_t sz(const T& v) {
  return std::size(v);
}

template <typename T>
int64_t popcount(T i) {
  return std::bitset<std::numeric_limits<T>::digits>(i).count();
}

template <typename T>
bool hasbit(T s, int i) {
  return std::bitset<std::numeric_limits<T>::digits>(s)[i];
}

template <typename T, typename U>
auto div_floor(T n, U d) {
  if (d < 0) {
    n = -n;
    d = -d;
  }
  if (n < 0) {
    return -((-n + d - 1) / d);
  }
  return n / d;
};

template <typename T, typename U>
auto div_ceil(T n, U d) {
  if (d < 0) {
    n = -n;
    d = -d;
  }
  if (n < 0) {
    return -(-n / d);
  }
  return (n + d - 1) / d;
}

template <typename T>
bool even(T x) {
  return x % 2 == 0;
}

std::array<std::pair<int64_t, int64_t>, 4> adjacent(int64_t i, int64_t j) {
  return {{{i + 1, j}, {i, j + 1}, {i - 1, j}, {i, j - 1}}};
}

bool inside(int64_t i, int64_t j, int64_t I, int64_t J) {
  return 0 <= i && i < I && 0 <= j && j < J;
}

template <typename T>
void sort(T& v) {
  return std::sort(v.begin(), v.end());
}

template <typename T, typename Compare>
void sort(T& v, Compare comp) {
  return std::sort(v.begin(), v.end(), comp);
}

template <typename T>
void reverse(T& v) {
  return std::reverse(v.begin(), v.end());
}

template <typename T>
typename T::value_type accumulate(const T& v) {
  return std::accumulate(v.begin(), v.end(), typename T::value_type());
}

// big = 2305843009213693951 = 2^61-1 ~= 2.3*10^18
const int64_t big = std::numeric_limits<int64_t>::max() / 4;

using i64 = int64_t;
using i32 = int32_t;

template <typename T>
using low_priority_queue =
    std::priority_queue<T, std::vector<T>, std::greater<T>>;

template <typename T>
using V = std::vector<T>;
template <typename T>
using VV = V<V<T>>;

void Main();

int main() {
  std::ios_base::sync_with_stdio(false);
  std::cin.tie(NULL);
  std::cout << std::fixed << std::setprecision(20);
  Main();
  return 0;
}

const auto& Fix = boost::hana::fix;

using namespace std;

#define int i64

using mint = ModInt<>;

using S = pair<mint, mint>;
S op(S a, S b) { return {a.first + b.first, a.second + b.second}; };
S e() { return {0, 0}; }
using F = mint;
S mapping(F f, S x) { return {x.first + f * x.second, x.second}; }
F composition(F f, F g) { return f + g; }
F id() { return 0; }
using segtree = atcoder::lazy_segtree<S, op, e, F, mapping, composition, id>;

void Main() {
  ints(n);
  V<int> s(n), c(n);
  cin >> s >> c;
  Graph<int> g(n);
  rep(n - 1) {
    ints(a, b);
    --a, --b;
    g.AddEdge(a, b);
    g.AddEdge(b, a);
  }
  HeavyLightDecomposition hld(g);
  segtree t(n);
  rep(i, n) t.set(hld.Index(i), {s[i], c[i]});
  ints(q);
  rep(q) {
    ints(k);
    if (k == 0) {
      ints(x, y, z);
      --x, --y;
      each(l, r, hld.Query(x, y, true)) t.apply(l, r, z);
    } else {
      ints(x, y);
      --x, --y;
      mint ans = 0;
      each(l, r, hld.Query(x, y, true)) ans += t.prod(l, r).first;
      wt(ans);
    }
  }
}