#include <bits/stdc++.h>
#pragma region Header
using i32 = int;
using u32 = unsigned int;
using i64 = long long;
using u64 = unsigned long long;
using i128 = __int128_t;
using u128 = __uint128_t;
using f64 = double;
using f80 = long double;
using f128 = __float128;
constexpr i32 operator"" _i32(u64 v)
{
    return v;
}
constexpr i32 operator"" _u32(u64 v)
{
    return v;
}
constexpr i64 operator"" _i64(u64 v)
{
    return v;
}
constexpr u64 operator"" _u64(u64 v)
{
    return v;
}
constexpr f64 operator"" _f64(f80 v)
{
    return v;
}
constexpr f80 operator"" _f80(f80 v)
{
    return v;
}
using Istream = std::istream;
using Ostream = std::ostream;
using Str = std::string;
template<typename T>
using Lt = std::less<T>;
template<typename T>
using Gt = std::greater<T>;
template<typename T>
using IList = std::initializer_list<T>;
template<int n>
using BSet = std::bitset<n>;
template<typename T1, typename T2>
using Pair = std::pair<T1, T2>;
template<typename... Ts>
using Tup = std::tuple<Ts...>;
template<typename T, int N>
using Arr = std::array<T, N>;
template<typename... Ts>
using Deq = std::deque<Ts...>;
template<typename... Ts>
using Set = std::set<Ts...>;
template<typename... Ts>
using MSet = std::multiset<Ts...>;
template<typename... Ts>
using USet = std::unordered_set<Ts...>;
template<typename... Ts>
using UMSet = std::unordered_multiset<Ts...>;
template<typename... Ts>
using Map = std::map<Ts...>;
template<typename... Ts>
using MMap = std::multimap<Ts...>;
template<typename... Ts>
using UMap = std::unordered_map<Ts...>;
template<typename... Ts>
using UMMap = std::unordered_multimap<Ts...>;
template<typename... Ts>
using Vec = std::vector<Ts...>;
template<typename... Ts>
using Stack = std::stack<Ts...>;
template<typename... Ts>
using Queue = std::queue<Ts...>;
template<typename T>
using MaxHeap = std::priority_queue<T>;
template<typename T>
using MinHeap = std::priority_queue<T, Vec<T>, Gt<T>>;
using NSec = std::chrono::nanoseconds;
using USec = std::chrono::microseconds;
using MSec = std::chrono::milliseconds;
using Sec = std::chrono::seconds;
template<typename T>
constexpr T LIMMIN = std::numeric_limits<T>::min();
template<typename T>
constexpr T LIMMAX = std::numeric_limits<T>::max();
template<typename T>
constexpr T INF = (LIMMAX<T> - 1) / 2;
template<typename T>
constexpr T PI = T{3.141592653589793238462643383279502884};
template<typename T = u64>
constexpr T TEN(const int n)
{
    return n == 0 ? T{1} : TEN<T>(n - 1) * T{10};
}
Ostream& operator<<(Ostream& os, i128 v)
{
    bool minus = false;
    if (v < 0) { minus = true, v = -v; }
    Str ans;
    if (v == 0) { ans = "0"; }
    while (v) {
        ans.push_back('0' + v % 10), v /= 10;
    }
    std::reverse(ans.begin(), ans.end());
    return os << (minus ? "-" : "") << ans;
}
Ostream& operator<<(Ostream& os, u128 v)
{
    Str ans;
    if (v == 0) { ans = "0"; }
    while (v) {
        ans.push_back('0' + v % 10), v /= 10;
    }
    std::reverse(ans.begin(), ans.end());
    return os << ans;
}
template<typename T>
bool chmin(T& a, const T& b)
{
    if (a > b) {
        a = b;
        return true;
    } else {
        return false;
    }
}
template<typename T>
bool chmax(T& a, const T& b)
{
    if (a < b) {
        a = b;
        return true;
    } else {
        return false;
    }
}
template<typename T>
constexpr T floorDiv(T x, T y)
{
    if (y < T{}) { x = -x, y = -y; }
    return x >= T{} ? x / y : (x - y + 1) / y;
}
template<typename T>
constexpr T ceilDiv(T x, T y)
{
    if (y < T{}) { x = -x, y = -y; }
    return x >= T{} ? (x + y - 1) / y : x / y;
}
template<typename T, typename I>
constexpr T modPower(T v, I n, T mod)
{
    T ans = 1 % mod;
    for (; n > 0; n >>= 1, (v *= v) %= mod) {
        if (n % 2 == 1) { (ans *= v) %= mod; }
    }
    return ans;
}
template<typename T, typename I>
constexpr T power(T v, I n)
{
    T ans = 1;
    for (; n > 0; n >>= 1, v *= v) {
        if (n % 2 == 1) { ans *= v; }
    }
    return ans;
}
template<typename T, typename I>
constexpr T power(T v, I n, const T& e)
{
    T ans = e;
    for (; n > 0; n >>= 1, v *= v) {
        if (n % 2 == 1) { ans *= v; }
    }
    return ans;
}
template<typename T>
Vec<T> operator+=(Vec<T>& vs1, const Vec<T>& vs2)
{
    vs1.insert(vs1.end(), vs2.begin(), vs2.end());
    return vs1;
}
template<typename T>
Vec<T> operator+(const Vec<T>& vs1, const Vec<T>& vs2)
{
    auto vs = vs1;
    vs += vs2;
    return vs;
}
template<typename Vs, typename V>
void fillAll(Vs& arr, const V& v)
{
    if constexpr (std::is_convertible<V, Vs>::value) {
        arr = v;
    } else {
        for (auto& subarr : arr) {
            fillAll(subarr, v);
        }
    }
}
template<typename Vs>
void sortAll(Vs& vs)
{
    std::sort(std::begin(vs), std::end(vs));
}
template<typename Vs, typename C>
void sortAll(Vs& vs, C comp)
{
    std::sort(std::begin(vs), std::end(vs), comp);
}
template<typename Vs>
void reverseAll(Vs& vs)
{
    std::reverse(std::begin(vs), std::end(vs));
}
template<typename V, typename Vs>
V sumAll(const Vs& vs)
{
    if constexpr (std::is_convertible<Vs, V>::value) {
        return static_cast<V>(vs);
    } else {
        V ans = 0;
        for (const auto& v : vs) {
            ans += sumAll<V>(v);
        }
        return ans;
    }
}
template<typename Vs>
int minInd(const Vs& vs)
{
    return std::min_element(std::begin(vs), std::end(vs)) - std::begin(vs);
}
template<typename Vs>
int maxInd(const Vs& vs)
{
    return std::max_element(std::begin(vs), std::end(vs)) - std::begin(vs);
}
template<typename Vs, typename V>
int lbInd(const Vs& vs, const V& v)
{
    return std::lower_bound(std::begin(vs), std::end(vs), v) - std::begin(vs);
}
template<typename Vs, typename V>
int ubInd(const Vs& vs, const V& v)
{
    return std::upper_bound(std::begin(vs), std::end(vs), v) - std::begin(vs);
}
template<typename T, typename F>
Vec<T> genVec(int n, F gen)
{
    Vec<T> ans;
    std::generate_n(std::back_insert_iterator(ans), n, gen);
    return ans;
}
Vec<int> iotaVec(int n, int offset = 0)
{
    Vec<int> ans(n);
    std::iota(ans.begin(), ans.end(), offset);
    return ans;
}
constexpr int popcount(const u64 v)
{
    return v ? __builtin_popcountll(v) : 0;
}
constexpr int log2p1(const u64 v)
{
    return v ? 64 - __builtin_clzll(v) : 0;
}
constexpr int lsbp1(const u64 v)
{
    return __builtin_ffsll(v);
}
constexpr int clog(const u64 v)
{
    return v ? log2p1(v - 1) : 0;
}
constexpr u64 ceil2(const u64 v)
{
    const int l = clog(v);
    return (l == 64) ? 0_u64 : (1_u64 << l);
}
constexpr u64 floor2(const u64 v)
{
    return v ? (1_u64 << (log2p1(v) - 1)) : 0_u64;
}
constexpr bool ispow2(const u64 v)
{
    return (v > 0) and ((v & (v - 1)) == 0);
}
constexpr bool btest(const u64 mask, const int ind)
{
    return (mask >> ind) & 1_u64;
}
template<typename F>
struct Fix : F
{
    Fix(F&& f) : F{std::forward<F>(f)} {}
    template<typename... Args>
    auto operator()(Args&&... args) const
    {
        return F::operator()(*this, std::forward<Args>(args)...);
    }
};
class irange
{
private:
    struct itr
    {
        itr(i64 start = 0, i64 step = 1) : m_cnt{start}, m_step{step} {}
        bool operator!=(const itr& it) const
        {
            return m_cnt != it.m_cnt;
        }
        int operator*()
        {
            return m_cnt;
        }
        itr& operator++()
        {
            m_cnt += m_step;
            return *this;
        }
        i64 m_cnt, m_step;
    };
    i64 m_start, m_end, m_step;
public:
    irange(i64 start, i64 end, i64 step = 1)
    {
        assert(step != 0);
        const i64 d = std::abs(step);
        const i64 l = (step > 0 ? start : end);
        const i64 r = (step > 0 ? end : start);
        int n = (r - l) / d + ((r - l) % d ? 1 : 0);
        if (l >= r) { n = 0; }
        m_start = start;
        m_end = start + step * n;
        m_step = step;
    }
    itr begin() const
    {
        return itr{m_start, m_step};
    }
    itr end() const
    {
        return itr{m_end, m_step};
    }
};
irange rep(int end)
{
    return irange(0, end, 1);
}
irange per(int rend)
{
    return irange(rend - 1, -1, -1);
}
#pragma COMMENT("[REFS] Xoshiro: https://prng.di.unimi.it")
namespace xoshiro_impl {
u64 x;
u64 next()
{
    uint64_t z = (x += 0x9e3779b97f4a7c15);
    z = (z ^ (z >> 30)) * 0xbf58476d1ce4e5b9;
    z = (z ^ (z >> 27)) * 0x94d049bb133111eb;
    return z ^ (z >> 31);
}
} // namespace xoshiro_impl
class Xoshiro32
{
public:
    using result_type = u32;
    using T = result_type;
    Xoshiro32(T seed = 0)
    {
        xoshiro_impl::x = seed;
        s[0] = xoshiro_impl::next();
        s[1] = xoshiro_impl::next();
        s[2] = xoshiro_impl::next();
        s[3] = xoshiro_impl::next();
    }
    static constexpr T min()
    {
        return LIMMIN<T>;
    }
    static constexpr T max()
    {
        return LIMMAX<T>;
    }
    T operator()()
    {
        return next();
    }
private:
    static constexpr T rotl(const T x, int k)
    {
        return (x << k) | (x >> (32 - k));
    }
    T next()
    {
        const T ans = rotl(s[1] * 5, 7) * 9;
        const T t = s[1] << 9;
        s[2] ^= s[0];
        s[3] ^= s[1];
        s[1] ^= s[2];
        s[0] ^= s[3];
        s[2] ^= t;
        s[3] = rotl(s[3], 11);
        return ans;
    }
    T s[4];
};
class Xoshiro64
{
public:
    using result_type = u64;
    using T = result_type;
    Xoshiro64(T seed = 0)
    {
        xoshiro_impl::x = seed;
        s[0] = xoshiro_impl::next();
        s[1] = xoshiro_impl::next();
        s[2] = xoshiro_impl::next();
        s[3] = xoshiro_impl::next();
    }
    static constexpr T min()
    {
        return LIMMIN<T>;
    }
    static constexpr T max()
    {
        return LIMMAX<T>;
    }
    T operator()()
    {
        return next();
    }
private:
    static constexpr T rotl(const T x, int k)
    {
        return (x << k) | (x >> (64 - k));
    }
    T next()
    {
        const T ans = rotl(s[1] * 5, 7) * 9;
        const T t = s[1] << 17;
        s[2] ^= s[0];
        s[3] ^= s[1];
        s[1] ^= s[2];
        s[0] ^= s[3];
        s[2] ^= t;
        s[3] = rotl(s[3], 45);
        return ans;
    }
    T s[4];
};
template<typename Rng>
class RNG
{
public:
    using result_type = typename Rng::result_type;
    using T = result_type;
    static constexpr T min()
    {
        return Rng::min();
    }
    static constexpr T max()
    {
        return Rng::max();
    }
    RNG() : RNG(std::random_device{}()) {}
    RNG(T seed) : m_rng(seed) {}
    T operator()()
    {
        return m_rng();
    }
    template<typename T>
    T val(T min, T max)
    {
        return std::uniform_int_distribution<T>(min, max)(m_rng);
    }
    template<typename T>
    Pair<T, T> pair(T min, T max)
    {
        return std::minmax({val<T>(min, max), val<T>(min, max)});
    }
    template<typename T>
    Vec<T> vec(int n, T min, T max)
    {
        return genVec<T>(n, [&]() { return val<T>(min, max); });
    }
    template<typename T>
    Vec<Vec<T>> vvec(int n, int m, T min, T max)
    {
        return genVec<Vec<T>>(n, [&]() { return vec(m, min, max); });
    }
private:
    Rng m_rng;
};
RNG<std::mt19937> rng;
RNG<std::mt19937_64> rng64;
RNG<Xoshiro32> rng_xo;
RNG<Xoshiro64> rng_xo64;
class Scanner
{
public:
    Scanner(Istream& is = std::cin) : m_is{is}
    {
        m_is.tie(nullptr)->sync_with_stdio(false);
    }
    template<typename T>
    T val()
    {
        T v;
        return m_is >> v, v;
    }
    template<typename T>
    T val(T offset)
    {
        return val<T>() - offset;
    }
    template<typename T>
    Vec<T> vec(int n)
    {
        return genVec<T>(n, [&]() { return val<T>(); });
    }
    template<typename T>
    Vec<T> vec(int n, T offset)
    {
        return genVec<T>(n, [&]() { return val<T>(offset); });
    }
    template<typename T>
    Vec<Vec<T>> vvec(int n, int m)
    {
        return genVec<Vec<T>>(n, [&]() { return vec<T>(m); });
    }
    template<typename T>
    Vec<Vec<T>> vvec(int n, int m, const T offset)
    {
        return genVec<Vec<T>>(n, [&]() { return vec<T>(m, offset); });
    }
    template<typename... Args>
    auto tup()
    {
        return Tup<Args...>{val<Args>()...};
    }
    template<typename... Args>
    auto tup(const Args&... offsets)
    {
        return Tup<Args...>{val<Args>(offsets)...};
    }
private:
    Istream& m_is;
};
Scanner in;
class Printer
{
public:
    Printer(Ostream& os = std::cout) : m_os{os}
    {
        m_os << std::fixed << std::setprecision(15);
    }
    template<typename... Args>
    int operator()(const Args&... args)
    {
        dump(args...);
        return 0;
    }
    template<typename... Args>
    int ln(const Args&... args)
    {
        dump(args...), m_os << '\n';
        return 0;
    }
    template<typename... Args>
    int el(const Args&... args)
    {
        dump(args...), m_os << std::endl;
        return 0;
    }
private:
    template<typename T>
    void dump(const T& v)
    {
        m_os << v;
    }
    template<typename T>
    void dump(const Vec<T>& vs)
    {
        for (const int i : rep(vs.size())) {
            m_os << (i ? " " : ""), dump(vs[i]);
        }
    }
    template<typename T>
    void dump(const Vec<Vec<T>>& vss)
    {
        for (const int i : rep(vss.size())) {
            m_os << (i ? "\n" : ""), dump(vss[i]);
        }
    }
    template<typename T, typename... Ts>
    int dump(const T& v, const Ts&... args)
    {
        dump(v), m_os << ' ', dump(args...);
        return 0;
    }
    Ostream& m_os;
};
Printer out;
template<typename T, int n, int i = 0>
auto ndVec(int const (&szs)[n], const T x = T{})
{
    if constexpr (i == n) {
        return x;
    } else {
        return std::vector(szs[i], ndVec<T, n, i + 1>(szs, x));
    }
}
template<typename T, typename F>
T binSearch(T ng, T ok, F check)
{
    while (std::abs(ok - ng) > 1) {
        const T mid = (ok + ng) / 2;
        (check(mid) ? ok : ng) = mid;
    }
    return ok;
}
template<u32 mod_, u32 root_, u32 max2p_>
class modint
{
    template<typename U = u32&>
    static U modRef()
    {
        static u32 s_mod = 0;
        return s_mod;
    }
    template<typename U = u32&>
    static U rootRef()
    {
        static u32 s_root = 0;
        return s_root;
    }
    template<typename U = u32&>
    static U max2pRef()
    {
        static u32 s_max2p = 0;
        return s_max2p;
    }
public:
    template<typename U = const u32>
    static constexpr std::enable_if_t<mod_ != 0, U> mod()
    {
        return mod_;
    }
    template<typename U = const u32>
    static std::enable_if_t<mod_ == 0, U> mod()
    {
        return modRef();
    }
    template<typename U = const u32>
    static constexpr std::enable_if_t<mod_ != 0, U> root()
    {
        return root_;
    }
    template<typename U = const u32>
    static std::enable_if_t<mod_ == 0, U> root()
    {
        return rootRef();
    }
    template<typename U = const u32>
    static constexpr std::enable_if_t<mod_ != 0, U> max2p()
    {
        return max2p_;
    }
    template<typename U = const u32>
    static std::enable_if_t<mod_ == 0, U> max2p()
    {
        return max2pRef();
    }
    template<typename U = u32>
    static void setMod(std::enable_if_t<mod_ == 0, U> m)
    {
        modRef() = m;
    }
    template<typename U = u32>
    static void setRoot(std::enable_if_t<mod_ == 0, U> r)
    {
        rootRef() = r;
    }
    template<typename U = u32>
    static void setMax2p(std::enable_if_t<mod_ == 0, U> m)
    {
        max2pRef() = m;
    }
    constexpr modint() : m_val{0} {}
    constexpr modint(i64 v) : m_val{normll(v)} {}
    constexpr void setRaw(u32 v)
    {
        m_val = v;
    }
    constexpr modint operator-() const
    {
        return modint{0} - (*this);
    }
    constexpr modint& operator+=(const modint& m)
    {
        m_val = norm(m_val + m.val());
        return *this;
    }
    constexpr modint& operator-=(const modint& m)
    {
        m_val = norm(m_val + mod() - m.val());
        return *this;
    }
    constexpr modint& operator*=(const modint& m)
    {
        m_val = normll((i64)m_val * (i64)m.val() % (i64)mod());
        return *this;
    }
    constexpr modint& operator/=(const modint& m)
    {
        return *this *= m.inv();
    }
    constexpr modint operator+(const modint& m) const
    {
        auto v = *this;
        return v += m;
    }
    constexpr modint operator-(const modint& m) const
    {
        auto v = *this;
        return v -= m;
    }
    constexpr modint operator*(const modint& m) const
    {
        auto v = *this;
        return v *= m;
    }
    constexpr modint operator/(const modint& m) const
    {
        auto v = *this;
        return v /= m;
    }
    constexpr bool operator==(const modint& m) const
    {
        return m_val == m.val();
    }
    constexpr bool operator!=(const modint& m) const
    {
        return not(*this == m);
    }
    friend Istream& operator>>(Istream& is, modint& m)
    {
        i64 v;
        return is >> v, m = v, is;
    }
    friend Ostream& operator<<(Ostream& os, const modint& m)
    {
        return os << m.val();
    }
    constexpr u32 val() const
    {
        return m_val;
    }
    template<typename I>
    constexpr modint pow(I n) const
    {
        return power(*this, n);
    }
    constexpr modint inv() const
    {
        return pow(mod() - 2);
    }
    static modint sinv(u32 n)
    {
        static Vec<modint> is{1, 1};
        for (u32 i = (u32)is.size(); i <= n; i++) {
            is.push_back(-is[mod() % i] * (mod() / i));
        }
        return is[n];
    }
    static modint fact(u32 n)
    {
        static Vec<modint> fs{1, 1};
        for (u32 i = (u32)fs.size(); i <= n; i++) {
            fs.push_back(fs.back() * i);
        }
        return fs[n];
    }
    static modint ifact(u32 n)
    {
        static Vec<modint> ifs{1, 1};
        for (u32 i = (u32)ifs.size(); i <= n; i++) {
            ifs.push_back(ifs.back() * sinv(i));
        }
        return ifs[n];
    }
    static modint comb(int n, int k)
    {
        return k > n or k < 0 ? modint{0} : fact(n) * ifact(n - k) * ifact(k);
    }
private:
    static constexpr u32 norm(u32 x)
    {
        return x < mod() ? x : x - mod();
    }
    static constexpr u32 normll(i64 x)
    {
        return norm(u32(x % (i64)mod() + (i64)mod()));
    }
    u32 m_val;
};
using modint_1000000007 = modint<1000000007, 5, 1>;
using modint_998244353 = modint<998244353, 3, 23>;
template<int id>
using modint_dynamic = modint<0, 0, id>;
template<typename T = int>
class Graph
{
    struct Edge
    {
        Edge() = default;
        Edge(int i, int t, T c) : id{i}, to{t}, cost{c} {}
        int id;
        int to;
        T cost;
        operator int() const
        {
            return to;
        }
    };
public:
    Graph(int n) : m_v{n}, m_edges(n) {}
    void addEdge(int u, int v, bool bi = false)
    {
        assert(0 <= u and u < m_v);
        assert(0 <= v and v < m_v);
        m_edges[u].emplace_back(m_e, v, 1);
        if (bi) { m_edges[v].emplace_back(m_e, u, 1); }
        m_e++;
    }
    void addEdge(int u, int v, const T& c, bool bi = false)
    {
        assert(0 <= u and u < m_v);
        assert(0 <= v and v < m_v);
        m_edges[u].emplace_back(m_e, v, c);
        if (bi) { m_edges[v].emplace_back(m_e, u, c); }
        m_e++;
    }
    const Vec<Edge>& operator[](const int u) const
    {
        assert(0 <= u and u < m_v);
        return m_edges[u];
    }
    Vec<Edge>& operator[](const int u)
    {
        assert(0 <= u and u < m_v);
        return m_edges[u];
    }
    int v() const
    {
        return m_v;
    }
    int e() const
    {
        return m_e;
    }
    friend Ostream& operator<<(Ostream& os, const Graph& g)
    {
        for (int u : rep(g.v())) {
            for (const auto& [id, v, c] : g[u]) {
                os << "[" << id << "]: ";
                os << u << "->" << v << "(" << c << ")\n";
            }
        }
        return os;
    }
    Vec<T> sizes(int root = 0) const
    {
        const int N = v();
        assert(0 <= root and root < N);
        Vec<T> ss(N, 1);
        Fix([&](auto dfs, int u, int p) -> void {
            for (const auto& [id, v, c] : m_edges[u]) {
                static_cast<void>(id);
                if (v == p) { continue; }
                dfs(v, u);
                ss[u] += ss[v];
            }
        })(root, -1);
        return ss;
    }
    Vec<T> depths(int root = 0) const
    {
        const int N = v();
        assert(0 <= root and root < N);
        Vec<T> ds(N, 0);
        Fix([&](auto dfs, int u, int p) -> void {
            for (const auto& [id, v, c] : m_edges[u]) {
                static_cast<void>(id);
                if (v == p) { continue; }
                ds[v] = ds[u] + c;
                dfs(v, u);
            }
        })(root, -1);
        return ds;
    }
    Vec<int> parents(int root = 0) const
    {
        const int N = v();
        assert(0 <= root and root < N);
        Vec<int> ps(N, -1);
        Fix([&](auto dfs, int u, int p) -> void {
            for (const auto& [id, v, c] : m_edges[u]) {
                static_cast<void>(id);
                if (v == p) { continue; }
                ps[v] = u;
                dfs(v, u);
            }
        })(root, -1);
        return ps;
    }
private:
    int m_v;
    int m_e = 0;
    Vec<Vec<Edge>> m_edges;
};
template<typename T>
class CentroidDecomp
{
public:
    CentroidDecomp(const Graph<T>& g) : m_cs(g.v())
    {
        const int N = g.v();
        Vec<int> szs(N, 1);
        Vec<bool> used(N, false);
        auto sizeDfs = Fix([&](auto dfs, int u, int p) -> int {
            szs[u] = 1;
            for (int v : g[u]) {
                if (v == p or used[v]) { continue; }
                szs[u] += dfs(v, u);
            }
            return szs[u];
        });
        auto getCentor = Fix([&](auto dfs, int u, int p, int tot) -> int {
            for (int v : g[u]) {
                if (v == p or used[v]) { continue; }
                if (szs[v] * 2 > tot) { return dfs(v, u, tot); }
            }
            if (tot == N) { m_center = u; }
            return u;
        });
        Fix([&](auto dfs, int u, int pc) -> void {
            const int tot = sizeDfs(u, -1);
            const int c = getCentor(u, -1, tot);
            used[c] = true;
            if (pc != -1) { m_cs.addEdge(pc, c); }
            for (int v : g[c]) {
                if (not used[v]) { dfs(v, c); }
            }
        })(0, -1);
    }
    int center() const
    {
        return m_center;
    }
    const Graph<>& centers() const
    {
        return m_cs;
    }
private:
    int m_center;
    Graph<> m_cs;
};
template<typename mint>
class FPS : public Vec<mint>
{
public:
    using std::vector<mint>::vector;
    using std::vector<mint>::resize;
    FPS(const Vec<mint>& vs) : Vec<mint>{vs} {}
    int size() const
    {
        return Vec<mint>::size();
    }
    int deg() const
    {
        return size() - 1;
    }
    FPS low(int n) const
    {
        return FPS{this->begin(), this->begin() + std::min(n, size())};
    }
    FPS rev() const
    {
        FPS ans = *this;
        reverseAll(ans);
        return ans;
    }
    mint eval(const mint& x) const
    {
        mint ans = 0;
        mint power = 1;
        for (int i : rep(size())) {
            ans += power * (*this)[i];
            power *= x;
        }
        return ans;
    }
    mint& operator[](const int n)
    {
        if (deg() < n) { resize(n + 1); }
        return Vec<mint>::operator[](n);
    }
    const mint& operator[](const int n) const
    {
        return Vec<mint>::operator[](n);
    }
    template<typename I>
    mint at(const I n) const
    {
        return (n < size() ? (*this)[n] : mint{0});
    }
    FPS operator-() const
    {
        FPS ans = *this;
        for (auto& v : ans) {
            v = -v;
        }
        return ans;
    }
    FPS& operator+=(const FPS& f)
    {
        for (int i : rep(f.size())) {
            (*this)[i] += f[i];
        }
        return *this;
    }
    FPS& operator-=(const FPS& f)
    {
        for (int i : rep(f.size())) {
            (*this)[i] -= f[i];
        }
        return *this;
    }
    FPS& operator*=(const FPS& f)
    {
        return (*this) = (*this) * f;
    }
    FPS& operator<<=(const int s)
    {
        return *this = (*this << s);
    }
    FPS& operator>>=(const int s)
    {
        return *this = (*this >> s);
    }
    FPS operator+(const FPS& f) const
    {
        return FPS(*this) += f;
    }
    FPS operator-(const FPS& f) const
    {
        return FPS(*this) -= f;
    }
    FPS operator*(const FPS& f) const
    {
        return mult(f, size() + f.size() - 1);
    }
    FPS operator<<(const int s) const
    {
        FPS ans(size() + s);
        for (int i : rep(size())) {
            ans[i + s] = (*this)[i];
        }
        return ans;
    }
    FPS operator>>(const int s) const
    {
        FPS ans;
        for (int i : irange(s, size())) {
            ans[i - s] = (*this)[i];
        }
        return ans;
    }
    friend Ostream& operator<<(Ostream& os, const FPS& f)
    {
        return os << static_cast<Vec<mint>>(f);
    }
    FPS derivative() const
    {
        FPS ans;
        for (int i : irange(1, size())) {
            ans[i - 1] = (*this)[i] * i;
        }
        return ans;
    }
    FPS integral() const
    {
        FPS ans;
        for (int i : irange(1, size() + 1)) {
            ans[i] = (*this)[i - 1] * mint::sinv(i);
        }
        return ans;
    }
    FPS mult(const FPS& f, int sz) const
    {
        if (sz == 0) { return FPS{}; }
        const int N = std::min(size(), sz) + std::min(f.size(), sz) - 1;
        if (N < 10) {
            FPS ans;
            for (int i : rep(sz)) {
                for (int j : rep(sz)) {
                    if (i + j >= sz) { break; }
                    ans[i + j] += this->at(i) * f.at(j);
                }
            }
            return ans;
        }
        if (N <= (1 << mint::max2p())) {
            auto ans = conv<mint>(*this, f, sz);
            return ans;
        } else {
            const auto cs1 = conv<submint1>(*this, f, sz);
            const auto cs2 = conv<submint2>(*this, f, sz);
            const auto cs3 = conv<submint3>(*this, f, sz);
            FPS ans((int)cs1.size());
            for (int i : rep(cs1.size())) {
                ans[i] = restore(cs1[i].val(), cs2[i].val(), cs3[i].val());
            }
            return ans;
        }
    }
    FPS smult(int p, const mint a, int sz) // *(1+ax^p) (mod x^sz)
    {
        FPS ans = low(sz);
        for (int i = 0; i + p < sz; i++) {
            ans[i + p] += (*this)[i] * a;
        }
        return ans;
    }
    FPS sdiv(int p, const mint& a, int sz) // *(1+ax^p)^(-1) (mod x^sz)
    {
        FPS ans = low(sz);
        for (int i = 0; i + p < sz; i++) {
            ans[i + p] -= ans[i] * a;
        }
        return ans;
    }
    FPS inv(int sz) const
    {
        const int n = size();
        assert((*this)[0].val() != 0);
        const int N = n * 2 - 1;
        if (N <= (1 << mint::max2p())) {
            FPS r{(*this)[0].inv()};
            for (int lg = 0, m = 1; m < sz; m <<= 1, lg++) {
                FPS f{this->begin(), this->begin() + std::min(n, 2 * m)};
                FPS g = r;
                f.resize(2 * m), g.resize(2 * m);
                trans(f, lg + 1, false), trans(g, lg + 1, false);
                for (int i : rep(2 * m)) {
                    f[i] *= g[i];
                }
                trans(f, lg + 1, true);
                std::fill(f.begin(), f.begin() + m, 0);
                trans(f, lg + 1, false);
                for (int i : rep(2 * m)) {
                    f[i] *= g[i];
                }
                trans(f, lg + 1, true);
                for (int i = m; i < std::min(2 * m, sz); i++) {
                    r[i] = -f[i];
                }
            }
            return r;
        } else {
            FPS g{(*this)[0].inv()};
            for (int lg = 0, m = 1; m < sz; m <<= 1, lg++) {
                g = FPS{2} * g - this->mult(g.mult(g, 2 * m), 2 * m);
            }
            return g.low(sz);
        }
    }
    FPS log(const int sz) const
    {
        assert((*this)[0].val() == 1);
        auto ans = derivative().mult(inv(sz), sz).integral();
        ans.resize(sz);
        return ans;
    }
    FPS exp(const int sz) const
    {
        const int l = lsb(sz);
        if (l == -1) { return FPS{1}.low(sz); }
        assert((*this)[0].val() == 0);
        const int n = size();
        const int N = n * 2 - 1;
        if (N <= (1 << mint::max2p())) {
            FPS f = {1, (*this)[1]}, g{1}, G{1, 1};
            for (int m = 2, lg = 1; m < sz; m <<= 1, lg++) {
                auto F = f;
                F.resize(2 * m), trans(F, lg + 1, false);
                FPS z(m);
                for (int i : rep(m)) {
                    z[i] = F[i] * G[i];
                }
                trans(z, lg, true);
                std::fill(z.begin(), z.begin() + m / 2, 0);
                trans(z, lg, false);
                for (int i : rep(m)) {
                    z[i] *= G[i];
                }
                trans(z, lg, true);
                for (int i : irange(m / 2, m)) {
                    g[i] = -z[i];
                }
                G = g, G.resize(m * 2), trans(G, lg + 1, false);
                auto q = low(m).derivative();
                q.resize(m), trans(q, lg, false);
                for (int i : rep(m)) {
                    q[i] *= F[i];
                }
                trans(q, lg, true);
                const auto df = f.derivative();
                for (int i : rep(m - 1)) {
                    q[i] -= df[i];
                }
                q.resize(m * 2);
                for (int i : rep(m - 1)) {
                    q[m + i] = q[i], q[i] = 0;
                }
                trans(q, lg + 1, false);
                for (int i : rep(m * 2)) {
                    q[i] *= G[i];
                }
                trans(q, lg + 1, true);
                q.pop_back();
                q = q.integral();
                for (int i = m; i < std::min(size(), m * 2); i++) {
                    q[i] += (*this)[i];
                }
                std::fill(q.begin(), q.begin() + m, 0);
                trans(q, lg + 1, false);
                for (int i = 0; i < m * 2; i++) {
                    q[i] *= F[i];
                }
                trans(q, lg + 1, true);
                for (int i = m; i < 2 * m; i++) {
                    f[i] = q[i];
                }
            }
            return f.low(sz);
        } else {
            FPS f{1};
            for (int m = 1; m < sz; m <<= 1) {
                auto g = low(2 * m);
                g[0] += 1;
                f.resize(2 * m);
                g -= f.log(2 * m);
                g = f.mult(g, 2 * m);
                for (int i = m; i < std::min(2 * m, g.size()); i++) {
                    f[i] = g[i];
                }
            }
            return f.low(sz);
        }
    }
    template<typename I>
    FPS pow(I n) const
    {
        return pow(n, deg() * n + 1);
    }
    template<typename I>
    FPS pow(I n, int sz) const
    {
        if (n == 0) { return FPS{1}.low(sz); }
        if (size() == 0) { return FPS{}; }
        const int p = lsb(deg() / n);
        if (p == -1) { return FPS{}; }
        const mint a = (*this)[p];
        FPS f = (*this) >> p;
        for (auto& c : f) {
            c /= a;
        }
        f = f.log(sz - p * n);
        for (auto& c : f) {
            c *= n;
        }
        f = f.exp(sz - p * n);
        FPS g;
        for (int i : rep(f.size())) {
            g[i + p * n] = f[i] * a.pow(n);
        }
        return g;
    }
    FPS tshift(const mint& c) const
    {
        const int N = size();
        FPS f(N), d(N);
        for (int i = 0; i < N; i++) {
            d[i] = c.pow(N - 1 - i) * mint::ifact(N - 1 - i);
        }
        for (int i = 0; i < N; i++) {
            f[i] = (*this)[i] * mint::fact(i);
        }
        f = f * d;
        FPS g(N);
        for (int i = 0; i < N; i++) {
            g[i] = f[i + N - 1] * mint::ifact(i);
        }
        return g;
    }
    FPS quot(const FPS& g) const
    {
        const int N = size(), M = g.size();
        if (N < M) { return FPS{}; }
        const auto fR = rev(), gR = g.rev();
        return fR.mult(gR.inv(N - M + 1), N - M + 1).rev();
    }
    FPS rem(const FPS& g) const
    {
        return (*this) - g * quot(g);
    }
private:
    int lsb() const
    {
        return lsb(deg());
    }
    int lsb(int sz) const
    {
        for (int p : rep(sz + 1)) {
            if ((*this)[p].val() != 0) { return p; }
        }
        return -1;
    }
    using submint1 = modint<469762049, 3, 26>;
    using submint2 = modint<167772161, 3, 25>;
    using submint3 = modint<754974721, 11, 24>;
    template<typename submint>
    static void trans(Vec<submint>& as, int lg, bool rev)
    {
        const int N = 1 << lg;
        assert((int)as.size() == N);
        Vec<submint> rs, irs;
        if (rs.empty()) {
            const submint r = submint(submint::root()), ir = r.inv();
            rs.resize(submint::max2p() + 1), irs.resize(submint::max2p() + 1);
            rs.back() = -r.pow((submint::mod() - 1) >> submint::max2p()),
            irs.back() = -ir.pow((submint::mod() - 1) >> submint::max2p());
            for (u32 i : irange(submint::max2p(), 0, -1)) {
                rs[i - 1] = -(rs[i] * rs[i]);
                irs[i - 1] = -(irs[i] * irs[i]);
            }
        }
        const auto drange = (rev ? irange(0, lg, 1) : irange(lg - 1, -1, -1));
        for (const int d : drange) {
            const int width = 1 << d;
            submint e = 1;
            for (int i = 0, j = 1; i < N; i += width * 2, j++) {
                for (int l = i, r = i + width; l < i + width; l++, r++) {
                    if (rev) {
                        const submint x = as[l], y = as[r];
                        as[l] = x + y, as[r] = (x - y) * e;
                    } else {
                        const submint x = as[l], y = as[r] * e;
                        as[l] = x + y, as[r] = x - y;
                    }
                }
                e *= (rev ? irs : rs)[lsbp1(j) + 1];
            }
        }
        if (rev) {
            const submint iN = submint{N}.inv();
            for (auto& a : as) {
                a *= iN;
            }
        }
    }
    template<typename submint>
    static Vec<submint> conv(const Vec<mint>& as, const Vec<mint>& bs, int sz)
    {
        const int an = std::min((int)as.size(), sz);
        const int bn = std::min((int)bs.size(), sz);
        const int M = an + bn - 1;
        const int lg = clog(M);
        const int L = 1 << lg;
        Vec<submint> As(L), Bs(L);
        for (int i : rep(an)) {
            As[i] = as[i].val();
        }
        for (int i : rep(bn)) {
            Bs[i] = bs[i].val();
        }
        trans(As, lg, false), trans(Bs, lg, false);
        for (int i : rep(L)) {
            As[i] *= Bs[i];
        }
        trans(As, lg, true);
        const int N = std::min(sz, (int)as.size() + (int)bs.size() - 1);
        As.resize(N);
        return As;
    }
    static constexpr submint2 ip1 = submint2{submint1::mod()}.inv();
    static constexpr submint3 ip2 = submint3{submint2::mod()}.inv();
    static constexpr submint3 ip1p2 = submint3{submint1::mod()}.inv() * ip2;
    static constexpr mint p1()
    {
        return mint{submint1::mod()};
    }
    static constexpr mint p1p2()
    {
        return p1() * mint{submint2::mod()};
    }
    static constexpr mint restore(int x1, int x2, int x3)
    {
        const int k0 = x1;
        const int k1 = (ip1 * (x2 - k0)).val();
        const int k2 = (ip1p2 * (x3 - k0) - ip2 * k1).val();
        return p1p2() * k2 + p1() * k1 + k0;
    }
};
#pragma endregion
int main()
{
    using mint = modint_998244353;
    const auto N = in.val<int>();
    const auto Qs = in.vec<mint>(N);
    Graph g(N);
    for (int i : rep(N - 1)) {
        const auto [u, v] = in.tup<int, int>(1, 1);
        g.addEdge(u, v, true);
    }
    const auto decomp = CentroidDecomp(g);
    const int C = decomp.center();
    const auto cg = decomp.centers();
    void(0);
    Vec<mint> ans(N, 0);
    Vec<bool> used(N, false);
    FPS<mint> Ss(N, mint::fact(N) * mint::fact(N));
    for (int i : rep(N)) {
        Ss[i] /= mint(i + 1) * (i + 1);
    }
    Fix([&](auto dfs, int c) -> void {
        used[c] = true;
        Vec<int> nus;
        for (int nu : g[c]) {
            if (used[nu]) { continue; }
            nus.push_back(nu);
        }
        // qs[d] = cから見て深さdの頂点の電荷総和
        FPS<mint> qs;
        // qss[i][d] = cから見て深さdの頂点(\in S_i)の電荷総和
        Vec<FPS<mint>> qss(nus.size());
        // uss[i] = S_iの頂点集合
        Vec<Vec<int>> uss(nus.size());
        // lss[i][j] = S_i[j]の深さ
        Vec<Vec<int>> lss(nus.size());
        for (int i : rep(nus.size())) {
            Fix([&](auto dfs, int i, int u, int p, int l) -> void {
                //const int l = dist(c, u);
                lss[i].push_back(l);
                uss[i].push_back(u);
                qs[l] += Qs[u];
                qss[i][l] += Qs[u];
                for (int v : g[u]) {
                    if (v == p or used[v]) { continue; }
                    dfs(i, v, u, l + 1);
                }
            })(i, nus[i], -1, 1);
        }
        qs[0] += Qs[c];
        qs = qs.rev();
        for (auto& qs : qss) {
            qs = qs.rev();
        }
        const FPS<mint> rs
            = qs
              * FPS<mint>(Ss.begin(),
                          Ss.begin() + std::min(2 * (int)qs.size() - 1, N));
        Vec<FPS<mint>> rss(nus.size());
        for (int i : rep(nus.size())) {
            rss[i] = qss[i]
                     * FPS<mint>(Ss.begin(),
                                 Ss.begin()
                                     + std::min(2 * (int)qss[i].size() - 1, N));
        }
        for (int i : rep(nus.size())) {
            for (int j : rep(uss[i].size())) {
                const int u = uss[i][j];
                const int l = lss[i][j];
                ans[u] += rs.at(qs.size() + l - 1);
                ans[u] -= rss[i].at(qss[i].size() + l - 1);
            }
        }
        ans[c] += rs[qs.size() - 1];
        for (int nc : cg[c]) {
            dfs(nc);
        }
    })(C);
    for (int i : rep(N)) {
        out.ln(ans[i]);
    }
    return 0;
}