#include <bits/stdc++.h>
namespace suisen {
    template <class T> bool chmin(T& x, const T& y) { return y >= x ? false : (x = y, true); }
    template <class T> bool chmax(T& x, const T& y) { return y <= x ? false : (x = y, true); }
    template <class T> constexpr int pow_m1(T n) { return -(n & 1) | 1; }
    template <class T> constexpr T fld(const T x, const T y) { T q = x / y, r = x % y; return q - ((x ^ y) < 0 and (r != 0)); }
    template <class T> constexpr T cld(const T x, const T y) { T q = x / y, r = x % y; return q + ((x ^ y) > 0 and (r != 0)); }
}
namespace suisen::macro {
#define IMPL_REPITER(cond) auto& begin() { return *this; } auto end() { return nullptr; } auto& operator*() { return _val; } auto& operator++() { return _val += _step, *this; } bool operator!=(std::nullptr_t) { return cond; }
    template <class Int, class IntL = Int, class IntStep = Int, std::enable_if_t<(std::is_signed_v<Int> == std::is_signed_v<IntL>), std::nullptr_t> = nullptr> struct rep_impl {
        Int _val; const Int _end, _step;
        rep_impl(Int n) : rep_impl(0, n) {}
        rep_impl(IntL l, Int r, IntStep step = 1) : _val(l), _end(r), _step(step) {}
        IMPL_REPITER((_val < _end))
    };
    template <class Int, class IntL = Int, class IntStep = Int, std::enable_if_t<(std::is_signed_v<Int> == std::is_signed_v<IntL>), std::nullptr_t> = nullptr> struct rrep_impl {
        Int _val; const Int _end, _step;
        rrep_impl(Int n) : rrep_impl(0, n) {}
        rrep_impl(IntL l, Int r) : _val(r - 1), _end(l), _step(-1) {}
        rrep_impl(IntL l, Int r, IntStep step) : _val(l + fld<Int>(r - l - 1, step) * step), _end(l), _step(-step) {}
        IMPL_REPITER((_val >= _end))
    };
    template <class Int, class IntStep = Int> struct repinf_impl {
        Int _val; const Int _step;
        repinf_impl(Int l, IntStep step = 1) : _val(l), _step(step) {}
        IMPL_REPITER((true))
    };
#undef IMPL_REPITER
}

#include <iostream>

#include <limits>
#include <type_traits>

namespace suisen {
    template <typename ...Constraints> using constraints_t = std::enable_if_t<std::conjunction_v<Constraints...>, std::nullptr_t>;

    template <typename T, typename = std::nullptr_t> struct bitnum { static constexpr int value = 0; };
    template <typename T> struct bitnum<T, constraints_t<std::is_integral<T>>> { static constexpr int value = std::numeric_limits<std::make_unsigned_t<T>>::digits; };
    template <typename T> static constexpr int bitnum_v = bitnum<T>::value;
    template <typename T, size_t n> struct is_nbit { static constexpr bool value = bitnum_v<T> == n; };
    template <typename T, size_t n> static constexpr bool is_nbit_v = is_nbit<T, n>::value;

    template <typename T, typename = std::nullptr_t> struct safely_multipliable { using type = T; };
    template <typename T> struct safely_multipliable<T, constraints_t<std::is_signed<T>, is_nbit<T, 32>>> { using type = long long; };
    template <typename T> struct safely_multipliable<T, constraints_t<std::is_signed<T>, is_nbit<T, 64>>> { using type = __int128_t; };
    template <typename T> struct safely_multipliable<T, constraints_t<std::is_unsigned<T>, is_nbit<T, 32>>> { using type = unsigned long long; };
    template <typename T> struct safely_multipliable<T, constraints_t<std::is_unsigned<T>, is_nbit<T, 64>>> { using type = __uint128_t; };
    template <typename T> using safely_multipliable_t = typename safely_multipliable<T>::type;

    template <typename T, typename = void> struct rec_value_type { using type = T; };
    template <typename T> struct rec_value_type<T, std::void_t<typename T::value_type>> {
        using type = typename rec_value_type<typename T::value_type>::type;
    };
    template <typename T> using rec_value_type_t = typename rec_value_type<T>::type;

    template <typename T> class is_iterable {
        template <typename T_> static auto test(T_ e) -> decltype(e.begin(), e.end(), std::true_type{});
        static std::false_type test(...);
    public:
        static constexpr bool value = decltype(test(std::declval<T>()))::value;
    };
    template <typename T> static constexpr bool is_iterable_v = is_iterable<T>::value;
    template <typename T> class is_writable {
        template <typename T_> static auto test(T_ e) -> decltype(std::declval<std::ostream&>() << e, std::true_type{});
        static std::false_type test(...);
    public:
        static constexpr bool value = decltype(test(std::declval<T>()))::value;
    };
    template <typename T> static constexpr bool is_writable_v = is_writable<T>::value;
    template <typename T> class is_readable {
        template <typename T_> static auto test(T_ e) -> decltype(std::declval<std::istream&>() >> e, std::true_type{});
        static std::false_type test(...);
    public:
        static constexpr bool value = decltype(test(std::declval<T>()))::value;
    };
    template <typename T> static constexpr bool is_readable_v = is_readable<T>::value;
} // namespace suisen
namespace suisen::io {
    template <typename IStream, std::enable_if_t<std::conjunction_v<std::is_base_of<std::istream, std::remove_reference_t<IStream>>, std::negation<std::is_const<std::remove_reference_t<IStream>>>>, std::nullptr_t> = nullptr>
    struct InputStream {
    private:
        using istream_type = std::remove_reference_t<IStream>;
        IStream is;
        struct { InputStream* is; template <typename T> operator T() { T e; *is >> e; return e; } } _reader{ this };
    public:
        template <typename IStream_> InputStream(IStream_ &&is) : is(std::move(is)) {}
        template <typename IStream_> InputStream(IStream_ &is) : is(is) {}
        template <typename T> InputStream& operator>>(T& e) {
            if constexpr (suisen::is_readable_v<T>) is >> e; else _read(e);
            return *this;
        }
        auto read() { return _reader; }
        template <typename Head, typename... Tail>
        void read(Head& head, Tail &...tails) { ((*this >> head) >> ... >> tails); }
        istream_type& get_stream() { return is; }
    private:
        static __uint128_t _stou128(const std::string& s) {
            __uint128_t ret = 0;
            for (char c : s) if ('0' <= c and c <= '9') ret = 10 * ret + c - '0';
            return ret;
        }
        static __int128_t _stoi128(const std::string& s) { return (s[0] == '-' ? -1 : +1) * _stou128(s); }

        void _read(__uint128_t& v) { v = _stou128(std::string(_reader)); }
        void _read(__int128_t& v) { v = _stoi128(std::string(_reader)); }
        template <typename T, typename U>
        void _read(std::pair<T, U>& a) { *this >> a.first >> a.second; }
        template <size_t N = 0, typename ...Args>
        void _read(std::tuple<Args...>& a) { if constexpr (N < sizeof...(Args)) *this >> std::get<N>(a), _read<N + 1>(a); }
        template <typename Iterable, std::enable_if_t<suisen::is_iterable_v<Iterable>, std::nullptr_t> = nullptr>
        void _read(Iterable& a) { for (auto& e : a) *this >> e; }
    };
    template <typename IStream>
    InputStream(IStream &&) -> InputStream<IStream>;
    template <typename IStream>
    InputStream(IStream &) -> InputStream<IStream&>;

    InputStream cin{ std::cin };

    auto read() { return cin.read(); }
    template <typename Head, typename... Tail>
    void read(Head& head, Tail &...tails) { cin.read(head, tails...); }
} // namespace suisen::io
namespace suisen { using io::read; } // namespace suisen

namespace suisen::io {
    template <typename OStream, std::enable_if_t<std::conjunction_v<std::is_base_of<std::ostream, std::remove_reference_t<OStream>>, std::negation<std::is_const<std::remove_reference_t<OStream>>>>, std::nullptr_t> = nullptr>
    struct OutputStream {
    private:
        using ostream_type = std::remove_reference_t<OStream>;
        OStream os;
    public:
        template <typename OStream_> OutputStream(OStream_ &&os) : os(std::move(os)) {}
        template <typename OStream_> OutputStream(OStream_ &os) : os(os) {}
        template <typename T> OutputStream& operator<<(const T& e) {
            if constexpr (suisen::is_writable_v<T>) os << e; else _print(e);
            return *this;
        }
        void print() { *this << '\n'; }
        template <typename Head, typename... Tail>
        void print(const Head& head, const Tail &...tails) { *this << head, ((*this << ' ' << tails), ...), *this << '\n'; }
        template <typename Iterable, std::enable_if_t<suisen::is_iterable_v<Iterable>, std::nullptr_t> = nullptr>
        void print_all(const Iterable& v, std::string sep = " ", std::string end = "\n") {
            for (auto it = v.begin(); it != v.end();) if (*this << *it; ++it != v.end()) *this << sep;
            *this << end;
        }
        ostream_type& get_stream() { return os; }
    private:
        void _print(__uint128_t value) {
            char buffer[41], *d = std::end(buffer);
            do *--d = '0' + (value % 10), value /= 10; while (value);
            os.rdbuf()->sputn(d, std::end(buffer) - d);
        }
        void _print(__int128_t value) {
            if (value < 0) *this << '-';
            _print(__uint128_t(value < 0 ? -value : value));
        }
        template <typename T, typename U>
        void _print(const std::pair<T, U>& a) { *this << a.first << ' ' << a.second; }
        template <size_t N = 0, typename ...Args>
        void _print(const std::tuple<Args...>& a) {
            if constexpr (N < std::tuple_size_v<std::tuple<Args...>>) {
                if constexpr (N) *this << ' ';
                *this << std::get<N>(a), _print<N + 1>(a);
            }
        }
        template <typename Iterable, std::enable_if_t<suisen::is_iterable_v<Iterable>, std::nullptr_t> = nullptr>
        void _print(const Iterable& a) { print_all(a, " ", ""); }
    };
    template <typename OStream_>
    OutputStream(OStream_ &&) -> OutputStream<OStream_>;
    template <typename OStream_>
    OutputStream(OStream_ &) -> OutputStream<OStream_&>;

    OutputStream cout{ std::cout }, cerr{ std::cerr };

    template <typename... Args>
    void print(const Args &... args) { cout.print(args...); }
    template <typename Iterable, std::enable_if_t<suisen::is_iterable_v<Iterable>, std::nullptr_t> = nullptr>
    void print_all(const Iterable& v, const std::string& sep = " ", const std::string& end = "\n") { cout.print_all(v, sep, end); }
} // namespace suisen::io
namespace suisen { using io::print, io::print_all; } // namespace suisen

namespace suisen {
    template <class T, class ToKey, class CompKey = std::less<>, std::enable_if_t<std::conjunction_v<std::is_invocable<ToKey, T>, std::is_invocable_r<bool, CompKey, std::invoke_result_t<ToKey, T>, std::invoke_result_t<ToKey, T>>>, std::nullptr_t> = nullptr>
    auto comparator(const ToKey& to_key, const CompKey& comp_key = std::less<>()) {
        return [=](const T& x, const T& y) { return comp_key(to_key(x), to_key(y)); };
    }
    template <class Compare, std::enable_if_t<std::is_invocable_r_v<bool, Compare, int, int>, std::nullptr_t> = nullptr>
    std::vector<int> sorted_indices(int n, const Compare& compare) {
        std::vector<int> p(n);
        return std::iota(p.begin(), p.end(), 0), std::sort(p.begin(), p.end(), compare), p;
    }
    template <class ToKey, std::enable_if_t<std::is_invocable_v<ToKey, int>, std::nullptr_t> = nullptr>
    std::vector<int> sorted_indices(int n, const ToKey& to_key) { return sorted_indices(n, comparator<int>(to_key)); }
    template <class T, class Comparator>
    auto priority_queue_with_comparator(const Comparator& comparator) { return std::priority_queue<T, std::vector<T>, Comparator>{ comparator }; }
    template <class Iterable, std::enable_if_t<suisen::is_iterable_v<Iterable>, std::nullptr_t> = nullptr>
    void sort_unique_erase(Iterable& a) { std::sort(a.begin(), a.end()), a.erase(std::unique(a.begin(), a.end()), a.end()); }

    template <size_t D> struct Dim : std::array<int, D> {
        template <typename ...Ints> Dim(const Ints& ...ns) : std::array<int, D>::array{ static_cast<int>(ns)... } {}
    };
    template <typename ...Ints> Dim(const Ints& ...) -> Dim<sizeof...(Ints)>;
    template <class T, size_t D, size_t I = 0>
    auto ndvec(const Dim<D> &ns, const T& value = {}) {
        if constexpr (I + 1 < D) {
            return std::vector(ns[I], ndvec<T, D, I + 1>(ns, value));
        } else {
            return std::vector<T>(ns[I], value);
        }
    }
}
namespace suisen {
    using int128 = __int128_t;
    using uint128 = __uint128_t;
    template <class T> using min_priority_queue = std::priority_queue<T, std::vector<T>, std::greater<T>>;
    template <class T> using max_priority_queue = std::priority_queue<T, std::vector<T>, std::less<T>>;
}
namespace suisen { const std::string Yes = "Yes", No = "No", YES = "YES", NO = "NO"; }

#ifdef LOCAL
#  define debug(...) debug_impl(#__VA_ARGS__, __VA_ARGS__)
template <class H, class... Ts> void debug_impl(const char* s, const H& h, const Ts&... t) {
    suisen::io::cerr << "[\033[32mDEBUG\033[m] " << s << ": " << h, ((suisen::io::cerr << ", " << t), ..., (suisen::io::cerr << "\n"));
}
#else
#  define debug(...) void(0)
#endif
#define FOR(e, v) for (auto &&e : v)
#define CFOR(e, v) for (const auto &e : v)
#define REP(i, ...) CFOR(i, suisen::macro::rep_impl(__VA_ARGS__))
#define RREP(i, ...) CFOR(i, suisen::macro::rrep_impl(__VA_ARGS__))
#define REPINF(i, ...) CFOR(i, suisen::macro::repinf_impl(__VA_ARGS__))
#define LOOP(n) for ([[maybe_unused]] const auto& _ : suisen::macro::rep_impl(n))
#define ALL(iterable) std::begin(iterable), std::end(iterable)

using namespace suisen;
using namespace std;
struct io_setup {
    io_setup(int precision = 20) {
        std::ios::sync_with_stdio(false), std::cin.tie(nullptr);
        std::cout << std::fixed << std::setprecision(precision);
    }
} io_setup_{};

constexpr int iinf = std::numeric_limits<int>::max() / 2;
constexpr long long linf = std::numeric_limits<long long>::max() / 2;

#include <deque>
#include <queue>
#include <tuple>
#include <vector>

#include <algorithm>
#include <cassert>
#include <optional>
#include <utility>

namespace suisen {
    namespace internal::csr_graph { struct graph_base_tag {}; }
    struct directed_graph_tag : internal::csr_graph::graph_base_tag {};
    struct undirected_graph_tag : internal::csr_graph::graph_base_tag {};
    template <typename T>
    struct is_graph_tag { static constexpr bool value = std::is_base_of_v<internal::csr_graph::graph_base_tag, T>; };
    template <typename T>
    constexpr bool is_graph_tag_v = is_graph_tag<T>::value;

    template <typename WeightType = void>
    struct Graph {
        template <typename GraphTag, typename, std::enable_if_t<is_graph_tag_v<GraphTag>, std::nullptr_t>>
        friend struct GraphBuilder;

        using weight_type = WeightType;
        static constexpr bool weighted = std::negation_v<std::is_same<weight_type, void>>;

        using weight_type_or_1 = std::conditional_t<weighted, weight_type, int>;

        using input_edge_type = std::conditional_t<weighted, std::tuple<int, int, weight_type>, std::pair<int, int>>;
    private:
        using internal_edge_type = std::conditional_t<weighted, std::pair<int, weight_type>, int>;
        struct Edge : public internal_edge_type {
            using internal_edge_type::internal_edge_type;
            operator int() const { return std::get<0>(*this); }
        };
    public:
        using edge_type = std::conditional_t<weighted, Edge, int>;
    private:
        struct AdjacentList {
            friend struct Graph;

            using value_type = edge_type;
            using iterator = typename std::vector<value_type>::iterator;
            using const_iterator = typename std::vector<value_type>::const_iterator;
            using reverse_iterator = typename std::vector<value_type>::reverse_iterator;
            using const_reverse_iterator = typename std::vector<value_type>::const_reverse_iterator;

            AdjacentList() = default;

            int size() const { return _siz; }
            bool empty() const { return _siz == 0; }
            int capacity() const { return _cap; }

            value_type& operator[](int i) { return *(begin() + i); }
            const value_type& operator[](int i) const { return *(cbegin() + i); }
            value_type& at(uint32_t i) { assert(i < _siz); return *(begin() + i); }
            const value_type& at(uint32_t i) const { assert(i < _siz); return *(cbegin() + i); }

            value_type* data() { return _g->_edges.data() + _offset; }
            const value_type* data() const { return _g->_edges.data() + _offset; }

            iterator begin() const { return _g->_edges.begin() + _offset; }
            iterator end() const { return begin() + _siz; }
            const_iterator cbegin() const { return _g->_edges.cbegin() + _offset; }
            const_iterator cend() const { return cbegin() + _siz; }
            reverse_iterator rbegin() const { return _g->_edges.rbegin() + (_g->_edges.size() - (_offset + _siz)); }
            reverse_iterator rend() const { return rbegin() + _siz; }
            const_reverse_iterator crbegin() const { return _g->_edges.crbegin() + (_g->_edges.size() - (_offset + _siz)); }
            const_reverse_iterator crend() const { return crbegin() + _siz; }

            void erase(const_iterator pos) {
                erase(pos, std::next(pos));
            }
            void erase(const_iterator first, const_iterator last) {
                const int num = last - first, k = first - cbegin();
                assert(num >= 0);
                if (num == 0) return;
                assert(0 <= k and k <= _siz - num);
                std::move(begin() + k + num, end(), begin() + k);
                _siz -= num;
            }
            void pop_back() {
                assert(_siz);
                --_siz;
            }
            void clear() { _siz = 0; }

            const value_type& back() const { return *--cend(); }
            value_type& back() { return *--end(); }
            const value_type& front() const { return *cbegin(); }
            value_type& front() { return *begin(); }

            void push_back(const value_type& x) {
                ++_siz;
                assert(_siz <= _cap);
                back() = x;
            }
            template <typename ...Args>
            void emplace_back(Args &&...args) {
                ++_siz;
                assert(_siz <= _cap);
                back() = value_type(std::forward<Args>(args)...);
            }

            void insert(const_iterator pos, const value_type& x) {
                emplace(pos, x);
            }
            void insert(const_iterator pos, int num, const value_type& x) {
                const int k = pos - cbegin();
                assert(0 <= k and k <= _siz);
                std::fill(begin() + k, shift_back(begin() + k, num), x);
            }
            template <class RandomAccessIterator>
            auto insert(const_iterator pos, RandomAccessIterator first, RandomAccessIterator last) -> decltype(*first++, last - first, void()) {
                const int num = last - first, k = pos - cbegin();
                assert(0 <= k and k <= _siz);
                shift_back(begin() + k, num);
                std::copy(first, last, begin() + k);
            }
            void insert(const_iterator pos, std::initializer_list<value_type> il) { insert(pos, il.begin(), il.end()); }
            template <typename ...Args>
            void emplace(const_iterator pos, Args &&...args) {
                const int k = pos - cbegin();
                assert(0 <= k and k <= _siz);
                *--shift_back(begin() + k) = value_type(std::forward<Args>(args)...);
            }
        private:
            mutable Graph* _g;
            int _cap;
            int _offset;
            int _siz;

            iterator shift_back(iterator pos, int num = 1) {
                _siz += num;
                assert(_siz <= _cap);
                return std::move_backward(pos, end() - num, end());
            }
        };
    public:
        using adjacent_list = AdjacentList;

        Graph() = default;

        template <typename GraphTag, std::enable_if_t<is_graph_tag_v<GraphTag>, std::nullptr_t> = nullptr>
        Graph(const int n, const std::vector<input_edge_type>& edges, GraphTag, std::vector<int> cap = {}) : _n(n), _adj(_n) {
            static constexpr bool undirected = std::is_same_v<undirected_graph_tag, GraphTag>;

            for (const auto& e : edges) {
                const int u = std::get<0>(e);
                ++_adj[u]._siz;
                if constexpr (undirected) {
                    const int v = std::get<1>(e);
                    ++_adj[v]._siz;
                }
            }
            if (cap.empty()) cap.resize(_n, std::numeric_limits<int>::max());
            int edge_num = 0;
            for (int i = 0; i < _n; ++i) {
                _adj[i]._g = this;
                _adj[i]._cap = std::min(_adj[i]._siz, cap[i]);
                _adj[i]._offset = edge_num;
                edge_num += _adj[i]._siz;
            }
            _edges.resize(edge_num);
            std::vector<typename std::vector<edge_type>::iterator> ptr(_n);
            for (int i = 0; i < _n; ++i) ptr[i] = _adj[i].begin();
            for (const auto& e : edges) {
                const int u = std::get<0>(e);
                const int v = std::get<1>(e);
                if constexpr (weighted) {
                    const weight_type& w = std::get<2>(e);
                    *ptr[u]++ = { v, w };
                    if constexpr (undirected) *ptr[v]++ = { u, w };
                } else {
                    *ptr[u]++ = v;
                    if constexpr (undirected) *ptr[v]++ = u;
                }
            }
        }
        Graph(const std::vector<std::vector<edge_type>>& g) : Graph(g.size(), make_edges(g), directed_graph_tag{}) {}

        static Graph create_directed_graph(const int n, const std::vector<input_edge_type>& edges, const std::vector<int>& cap = {}) {
            return Graph(n, edges, directed_graph_tag{}, cap);
        }
        static Graph create_undirected_graph(const int n, const std::vector<input_edge_type>& edges, const std::vector<int>& cap = {}) {
            return Graph(n, edges, undirected_graph_tag{}, cap);
        }

        adjacent_list& operator[](int i) {
            _adj[i]._g = this;
            return _adj[i];
        }
        const adjacent_list& operator[](int i) const {
            _adj[i]._g = const_cast<Graph*>(this);
            return _adj[i];
        }

        int size() const {
            return _n;
        }

        void shrink_to_fit() {
            int edge_num = 0;
            for (const auto& l : _adj) edge_num += l.size();

            std::vector<edge_type> new_edges(edge_num);
            auto it = new_edges.begin();
            for (int i = 0; i < _n; ++i) {
                int nl = it - new_edges.begin();
                it = std::move(_adj[i].begin(), _adj[i].end(), it);
                _adj[i]._offset = nl;
                _adj[i]._cap = _adj[i]._siz;
            }
            _edges.swap(new_edges);
        }

        static weight_type_or_1 get_weight(const edge_type& edge) {
            if constexpr (weighted) return std::get<1>(edge);
            else return 1;
        }

        Graph reversed(const std::vector<int>& cap = {}) const {
            std::vector<input_edge_type> edges;
            for (int i = 0; i < _n; ++i) {
                for (const auto& edge : (*this)[i]) {
                    if constexpr (weighted) edges.emplace_back(std::get<0>(edge), i, std::get<1>(edge));
                    else edges.emplace_back(edge, i);
                }
            }
            return Graph(_n, std::move(edges), directed_graph_tag{}, cap);
        }

        struct DFSTree {
            std::vector<int> par;
            std::vector<int> pre_ord, pst_ord;
            Graph tree, back;
        };

        DFSTree dfs_tree(int start = 0) const {
            std::vector<input_edge_type> tree_edge, back_edge;

            std::vector<int> pre(_n), pst(_n);
            auto pre_it = pre.begin(), pst_it = pst.begin();

            std::vector<int> eid(_n, -1), par(_n, -2);
            std::vector<std::optional<weight_type_or_1>> par_w(_n, std::nullopt);
            for (int i = 0; i < _n; ++i) {
                int cur = (start + i) % _n;
                if (par[cur] != -2) continue;
                par[cur] = -1;
                while (cur >= 0) {
                    ++eid[cur];
                    if (eid[cur] == 0) *pre_it++ = cur;
                    if (eid[cur] == _adj[cur].size()) {
                        *pst_it++ = cur;
                        cur = par[cur];
                    } else {
                        const auto &e = _adj[cur][eid[cur]];
                        weight_type_or_1 w = get_weight(e);
                        int nxt = e;
                        if (par[nxt] == -2) {
                            tree_edge.emplace_back(make_edge(cur, e));
                            par[nxt] = cur;
                            par_w[nxt] = std::move(w);
                            cur = nxt;
                        } else if (eid[nxt] != _adj[nxt].size()) {
                            if (par[cur] != nxt or par_w[cur] != w or not std::exchange(par_w[cur], std::nullopt).has_value()) {
                                back_edge.emplace_back(make_edge(cur, e));
                            }
                        }
                    }
                }
            }
            Graph tree = create_directed_graph(_n, tree_edge);
            Graph back = create_directed_graph(_n, back_edge);
            return DFSTree{ std::move(par), std::move(pre), std::move(pst), std::move(tree), std::move(back) };
        }

    private:
        int _n;
        std::vector<adjacent_list> _adj;
        std::vector<edge_type> _edges;

        static std::vector<input_edge_type> make_edges(const std::vector<std::vector<edge_type>>& g) {
            const int n = g.size();
            std::vector<input_edge_type> edges;
            for (int i = 0; i < n; ++i) for (const auto& e : g[i]) {
                edges.emplace_back(make_edge(i, e));
            }
            return edges;
        }
        static input_edge_type make_edge(int i, const edge_type& e) {
            if constexpr (weighted) return { i, std::get<0>(e), std::get<1>(e) };
            else return { i, e };
        }
    };

    template <typename GraphTag>
    Graph(int, std::vector<std::pair<int, int>>, GraphTag, std::vector<int> = {})->Graph<void>;
    template <typename WeightType, typename GraphTag>
    Graph(int, std::vector<std::tuple<int, int, WeightType>>, GraphTag, std::vector<int> = {})->Graph<WeightType>;

    Graph(std::vector<std::vector<int>>)->Graph<void>;
    template <typename WeightType>
    Graph(std::vector<std::vector<std::pair<int, WeightType>>>)->Graph<WeightType>;

    template <typename GraphTag, typename WeightType = void,
        std::enable_if_t<is_graph_tag_v<GraphTag>, std::nullptr_t> = nullptr>
    struct GraphBuilder {
        using graph_tag = GraphTag;
        using weight_type = WeightType;
        using edge_type = typename Graph<weight_type>::input_edge_type;

        GraphBuilder(int n = 0) : _n(n) {}

        void add_edge(const edge_type& edge) {
            check_not_moved();
            _edges.push_back(edge);
        }
        template <typename ...Args>
        void emplace_edge(Args &&...args) {
            check_not_moved();
            _edges.emplace_back(std::forward<Args>(args)...);
        }
        template <typename EdgeContainer, std::enable_if_t<std::is_constructible_v<edge_type, typename EdgeContainer::value_type>, std::nullptr_t> = nullptr>
        void add_edges(const EdgeContainer& edges) {
            for (const auto& edge : edges) add_edge(edge);
        }

        template <bool move_edges = true>
        Graph<weight_type> build() {
            if constexpr (move_edges) {
                _moved = true;
                return Graph<weight_type>(_n, std::move(_edges), graph_tag{});
            } else {
                return Graph<weight_type>(_n, _edges, graph_tag{});
            }
        }
        Graph<weight_type> build_without_move() {
            return build<false>();
        }

        static Graph<weight_type> build(const int n, const std::vector<edge_type>& edges) {
            GraphBuilder builder(n);
            builder.add_edges(edges);
            return builder.build();
        }
    private:
        int _n;
        std::vector<edge_type> _edges;
        bool _moved = false;

        void check_not_moved() {
            if (not _moved) return;
            std::cerr << "[\033[31mERROR\033[m] Edges are already moved. If you want to add edges after calling build() and build another graph, you should use build_without_move() instead." << std::endl;
            assert(false);
        }
    };
    template <typename WeightType = void>
    using DirectedGraphBuilder = GraphBuilder<directed_graph_tag, WeightType>;
    template <typename WeightType = void>
    using UndirectedGraphBuilder = GraphBuilder<undirected_graph_tag, WeightType>;

    template <typename Weight, std::enable_if_t<std::negation_v<std::is_same<Weight, void>>, std::nullptr_t> = nullptr>
    using WeightedGraph = Graph<Weight>;
    using UnweightedGraph = Graph<void>;

    template <typename T>
    struct is_weighted_graph { static constexpr bool value = false; };
    template <typename WeightType>
    struct is_weighted_graph<Graph<WeightType>> { static constexpr bool value = Graph<WeightType>::weighted; };
    template <typename T>
    constexpr bool is_weighted_graph_v = is_weighted_graph<T>::value;

    template <typename T>
    struct is_unweighted_graph { static constexpr bool value = false; };
    template <typename WeightType>
    struct is_unweighted_graph<Graph<WeightType>> { static constexpr bool value = not Graph<WeightType>::weighted; };
    template <typename T>
    constexpr bool is_unweighted_graph_v = is_unweighted_graph<T>::value;
} // namespace suisen

namespace suisen {
    namespace internal {
        template <typename WeightType = void>
        struct CentroidDecomposition : Graph<WeightType> {
            friend struct CentroidDecompositionUnweighted;
            template <typename WeightType_, std::enable_if_t<not std::is_same_v<WeightType_, void>, std::nullptr_t>>
            friend struct CentroidDecompositionWeighted;

            using graph_type = Graph<WeightType>;
            using weight_type = WeightType;

            CentroidDecomposition(const graph_type& g) : graph_type(g), n(this->size()), cpar(n, -1), cdep(n, std::numeric_limits<int>::max()), csiz(n) {
                build();
            }

            int dct_parent(int i) const { return cpar[i]; }
            int dct_depth(int i) const { return cdep[i]; }
            int dct_size(int i) const { return csiz[i]; }

            auto collect_adj(int root, int u) {
                using iterator = typename Graph<WeightType>::adjacent_list::iterator;
                struct iter {
                    iter(CentroidDecomposition* ptr, int root, int u) :
                        _ptr(ptr),
                        _dr(ptr->dct_depth(root)),
                        _cur((*ptr)[u].begin()), _end((*ptr)[u].end()) { _succ(); }

                    decltype(auto) operator*() { return *_cur; }
                    iter& operator++() {
                        ++_cur, _succ();
                        return *this;
                    }
                    bool operator!=(std::nullptr_t) { return _cur != _end; }
                    auto begin() { return *this; }
                    auto end() { return nullptr; }
                private:
                    CentroidDecomposition* _ptr;
                    int _dr;
                    iterator _cur, _end;

                    void _succ() { while (_cur != _end and _ptr->dct_depth(int(*_cur)) < _dr) ++_cur; }
                };
                return iter{ this, root, u };
            }

        private:
            int n;
            std::vector<int> cpar;
            std::vector<int> cdep;
            std::vector<int> csiz;

            void build() {
                std::vector<int> eid(n, 0);

                cpar[0] = -1, csiz[0] = n;
                std::deque<std::tuple<int, int>> dq{ { 0, 0 } };

                while (dq.size()) {
                    const auto [r, dep] = dq.front();
                    const int siz = csiz[r], prev_ctr = cpar[r];
                    dq.pop_front();

                    int c = -1;
                    eid[r] = 0, csiz[r] = 1, cpar[r] = -1;
                    for (int cur = r;;) {
                        for (const int edge_num = int((*this)[cur].size());;) {
                            if (eid[cur] == edge_num) {
                                if (csiz[cur] * 2 > siz) {
                                    c = cur;
                                } else {
                                    const int nxt = cpar[cur];
                                    csiz[nxt] += csiz[cur];
                                    cur = nxt;
                                }
                                break;
                            }
                            const int nxt = (*this)[cur][eid[cur]++];
                            if (cdep[nxt] >= dep and nxt != cpar[cur]) {
                                eid[nxt] = 0, csiz[nxt] = 1, cpar[nxt] = cur;
                                cur = nxt;
                                break;
                            }
                        }
                        if (c >= 0) break;
                    }
                    for (int v : (*this)[c]) if (cdep[v] >= dep) {
                        if (cpar[c] == v) cpar[v] = c, csiz[v] = siz - csiz[c];
                        dq.emplace_back(v, dep + 1);
                    }
                    cpar[c] = prev_ctr, cdep[c] = dep, csiz[c] = siz;
                }
            }
        };

        struct CentroidDecompositionUnweighted : internal::CentroidDecomposition<void> {
            using base_type = internal::CentroidDecomposition<void>;
            using base_type::base_type;

            std::vector<std::vector<std::pair<int, int>>> collect(int root, int root_val = 0) const {
                std::vector<std::vector<std::pair<int, int>>> res{ { { root, root_val } } };
                for (int sub_root : (*this)[root]) if (this->cdep[sub_root] > this->cdep[root]) {
                    res.emplace_back();
                    std::deque<std::tuple<int, int, int>> dq{ { sub_root, root, root_val + 1 } };
                    while (dq.size()) {
                        auto [u, p, w] = dq.front();
                        dq.pop_front();
                        res.back().emplace_back(u, w);
                        for (int v : (*this)[u]) if (v != p and this->cdep[v] > this->cdep[root]) {
                            dq.emplace_back(v, u, w + 1);
                        }
                    }
                    std::copy(res.back().begin(), res.back().end(), std::back_inserter(res.front()));
                }
                return res;
            }
        };

        template <typename WeightType, std::enable_if_t<not std::is_same_v<WeightType, void>, std::nullptr_t> = nullptr>
        struct CentroidDecompositionWeighted : internal::CentroidDecomposition<WeightType> {
            using base_type = internal::CentroidDecomposition<WeightType>;
            using base_type::base_type;

            using weight_type = typename base_type::weight_type;

            template <typename Op, std::enable_if_t<std::is_invocable_r_v<weight_type, Op, weight_type, weight_type>, std::nullptr_t> = nullptr>
            std::vector<std::vector<std::pair<int, weight_type>>> collect(int root, Op op, weight_type root_val) const {
                std::vector<std::vector<std::pair<int, weight_type>>> res{ { { root, root_val } } };
                for (auto [sub_root, ew] : (*this)[root]) if (this->cdep[sub_root] > this->cdep[root]) {
                    res.emplace_back();
                    std::deque<std::tuple<int, int, weight_type>> dq{ { sub_root, root, op(root_val, ew) } };
                    while (dq.size()) {
                        auto [u, p, w] = dq.front();
                        dq.pop_front();
                        res.back().emplace_back(u, w);
                        for (auto [v, ew] : (*this)[u]) if (v != p and this->cdep[v] > this->cdep[root]) {
                            dq.emplace_back(v, u, op(w, ew));
                        }
                    }
                    std::copy(res.back().begin(), res.back().end(), std::back_inserter(res.front()));
                }
                return res;
            }
        };
    }

    using CentroidDecompositionUnweighted = internal::CentroidDecompositionUnweighted;
    template <typename WeightType, std::enable_if_t<not std::is_same_v<WeightType, void>, std::nullptr_t> = nullptr>
    using CentroidDecompositionWeighted = internal::CentroidDecompositionWeighted<WeightType>;

} // namespace suisen

namespace suisen {
template <typename T>
class CoordinateCompressorBuilder {
    public:
        struct Compressor {
            public:
                static constexpr int absent = -1;

                // default constructor
                Compressor() : _xs(std::vector<T>{}) {}
                // Construct from strictly sorted vector
                Compressor(const std::vector<T> &xs) : _xs(xs) {
                    assert(is_strictly_sorted(xs));
                }

                // Return the number of distinct keys.
                int size() const {
                    return _xs.size();
                }
                // Check if the element is registered.
                bool has_key(const T &e) const {
                    return std::binary_search(_xs.begin(), _xs.end(), e);
                }
                // Compress the element. if not registered, returns `default_value`. (default: Compressor::absent)
                int comp(const T &e, int default_value = absent) const {
                    const int res = min_geq_index(e);
                    return res != size() and _xs[res] == e ? res : default_value;
                }
                // Restore the element from the index.
                T decomp(const int compressed_index) const {
                    return _xs[compressed_index];
                }
                // Compress the element. Equivalent to call `comp(e)`
                int operator[](const T &e) const {
                    return comp(e);
                }
                // Return the minimum registered value greater than `e`. if not exists, return `default_value`.
                T min_gt(const T &e, const T &default_value) const {
                    auto it = std::upper_bound(_xs.begin(), _xs.end(), e);
                    return it == _xs.end() ? default_value : *it;
                }
                // Return the minimum registered value greater than or equal to `e`. if not exists, return `default_value`.
                T min_geq(const T &e, const T &default_value) const {
                    auto it = std::lower_bound(_xs.begin(), _xs.end(), e);
                    return it == _xs.end() ? default_value : *it;
                }
                // Return the maximum registered value less than `e`. if not exists, return `default_value`
                T max_lt(const T &e, const T &default_value) const {
                    auto it = std::upper_bound(_xs.rbegin(), _xs.rend(), e, std::greater<T>());
                    return it == _xs.rend() ? default_value : *it;
                }
                // Return the maximum registered value less than or equal to `e`. if not exists, return `default_value`
                T max_leq(const T &e, const T &default_value) const {
                    auto it = std::lower_bound(_xs.rbegin(), _xs.rend(), e, std::greater<T>());
                    return it == _xs.rend() ? default_value : *it;
                }
                // Return the compressed index of the minimum registered value greater than `e`. if not exists, return `compressor.size()`.
                int min_gt_index(const T &e) const {
                    return std::upper_bound(_xs.begin(), _xs.end(), e) - _xs.begin();
                }
                // Return the compressed index of the minimum registered value greater than or equal to `e`. if not exists, return `compressor.size()`.
                int min_geq_index(const T &e) const {
                    return std::lower_bound(_xs.begin(), _xs.end(), e) - _xs.begin();
                }
                // Return the compressed index of the maximum registered value less than `e`. if not exists, return -1.
                int max_lt_index(const T &e) const {
                    return int(_xs.rend() - std::upper_bound(_xs.rbegin(), _xs.rend(), e, std::greater<T>())) - 1;
                }
                // Return the compressed index of the maximum registered value less than or equal to `e`. if not exists, return -1.
                int max_leq_index(const T &e) const {
                    return int(_xs.rend() - std::lower_bound(_xs.rbegin(), _xs.rend(), e, std::greater<T>())) - 1;
                }
            private:
                std::vector<T> _xs;
                static bool is_strictly_sorted(const std::vector<T> &v) {
                    return std::adjacent_find(v.begin(), v.end(), std::greater_equal<T>()) == v.end();
                }
        };
        CoordinateCompressorBuilder() : _xs(std::vector<T>{}) {}
        explicit CoordinateCompressorBuilder(const std::vector<T> &xs) : _xs(xs) {}
        explicit CoordinateCompressorBuilder(std::vector<T> &&xs) : _xs(std::move(xs)) {}
        template <typename Gen, constraints_t<std::is_invocable_r<T, Gen, int>> = nullptr>
        CoordinateCompressorBuilder(const int n, Gen generator) {
            reserve(n);
            for (int i = 0; i < n; ++i) push(generator(i));
        }
        // Attempt to preallocate enough memory for specified number of elements.
        void reserve(int n) {
            _xs.reserve(n);
        }
        // Add data.
        void push(const T &first) {
            _xs.push_back(first);
        }
        // Add data.
        void push(T &&first) {
            _xs.push_back(std::move(first));
        }
        // Add data in the range of [first, last). 
        template <typename Iterator>
        auto push(const Iterator &first, const Iterator &last) -> decltype(std::vector<T>{}.push_back(*first), void()) {
            for (auto it = first; it != last; ++it) _xs.push_back(*it);
        }
        // Add all data in the container. Equivalent to `push(iterable.begin(), iterable.end())`.
        template <typename Iterable>
        auto push(const Iterable &iterable) -> decltype(std::vector<T>{}.push_back(*iterable.begin()), void()) {
            push(iterable.begin(), iterable.end());
        }
        // Add data.
        template <typename ...Args>
        void emplace(Args &&...args) {
            _xs.emplace_back(std::forward<Args>(args)...);
        }
        // Build compressor.
        auto build() {
            std::sort(_xs.begin(), _xs.end()), _xs.erase(std::unique(_xs.begin(), _xs.end()), _xs.end());
            return Compressor {_xs};
        }
        // Build compressor from vector.
        static auto build(const std::vector<T> &xs) {
            return CoordinateCompressorBuilder(xs).build();
        }
        // Build compressor from vector.
        static auto build(std::vector<T> &&xs) {
            return CoordinateCompressorBuilder(std::move(xs)).build();
        }
        // Build compressor from generator.
        template <typename Gen, constraints_t<std::is_invocable_r<T, Gen, int>> = nullptr>
        static auto build(const int n, Gen generator) {
            return CoordinateCompressorBuilder<T>(n, generator).build();
        }
    private:
        std::vector<T> _xs;
};

} // namespace suisen

constexpr int INF = 1 << 30;

void solve() {
    int n;
    read(n);

    vector<int> a(n);
    read(a);

    UnweightedGraph g = [&] {
        std::vector<std::vector<int>> g(n);
        LOOP(n - 1) {
            int u, v;
            read(u, v);
            --u, --v;
            g[u].push_back(v);
            g[v].push_back(u);
        }
        return g;
    }();

    vector<int> comp_id(n);
    vector<int> dep(n);

    CentroidDecompositionUnweighted cd(g);

    vector<int> ans(n);
    for (int root = 0; root < n; ++root) {
        vector<pair<int, int>> vs;

        int c = 0;
        for (int subroot : cd.collect_adj(root, root)) {
            auto dfs = [&](auto dfs, int u, int p, int d) -> void {
                vs.emplace_back(a[u], u);
                comp_id[u] = c;
                dep[u] = d;
                for (int v : cd.collect_adj(root, u)) if (v != p) {
                    dfs(dfs, v, u, d + 1);
                }
            };
            dfs(dfs, subroot, root, 1);
            ++c;
        }
        vs.emplace_back(a[root], root);
        comp_id[root] = c;
        dep[root] = 0;

        array<array<int, 2>, 3> top;
        top.fill({ -INF, -1 });

        auto update_max = [&](int dep, int cid) {
            top[2] = { dep, cid };
            if (top[1][0] < top[2][0]) swap(top[1], top[2]);
            if (top[0][0] < top[1][0]) swap(top[0], top[1]);
            if (top[0][1] == top[1][1]) swap(top[1], top[2]);
        };
        auto get_max_dep = [&](int cid) {
            return top[0][1] != cid ? top[0][0] : top[1][0];
        };

        sort(ALL(vs), greater<>());

        const int siz = vs.size();
        for (int l = 0; l < siz;) {
            int r = l;
            while (r < siz and vs[l].first == vs[r].first) {
                int v = vs[r++].second;
                update_max(dep[v], comp_id[v]);
            }
            while (l < r) {
                int v = vs[l++].second;
                ans[v] = max(ans[v], dep[v] + get_max_dep(comp_id[v]));
            }
        }
    }
    print(ans);
}

int main() {
    int test_case_num = 1;
    // read(test_case_num);
    LOOP(test_case_num) solve();
    return 0;
}