Problem: Alice and Bob are playing a game. They have a tree consisting of n vertices. Initially, Bob has k chips, the i-th chip is located in the vertex a_i (all these vertices are unique). Before the game starts, Alice will place a chip into one of the vertices of the tree.

The game consists of turns. Each turn, the following events happen (sequentially, exactly in the following order):

  1. Alice either moves her chip to an adjacent vertex or doesn't move it; 
  2. for each Bob's chip, he either moves it to an adjacent vertex or doesn't move it. Note that this choice is done independently for each chip. 



The game ends when Alice's chip shares the same vertex with one (or multiple) of Bob's chips. Note that Bob's chips may share the same vertex, even though they are in different vertices at the beginning of the game.

Alice wants to maximize the number of turns, Bob wants to minimize it. If the game ends in the middle of some turn (Alice moves her chip to a vertex that contains one or multiple Bob's chips), this turn is counted.

For each vertex, calculate the number of turns the game will last if Alice places her chip in that vertex.

Input

The first line contains one integer n (2 ≤ n ≤ 2 ⋅ 10^5) — the number of vertices in the tree.

Then n - 1 lines follow, each line contains two integers u_i, v_i (1 ≤ u_i, v_i ≤ n; u_i ≠ v_i) that denote the endpoints of an edge. These edges form a tree.

The next line contains one integer k (1 ≤ k ≤ n - 1) — the number of Bob's chips.

The last line contains k integers a_1, a_2, ..., a_k (1 ≤ a_i ≤ n; a_i ≠ a_j if i ≠ j) — the vertices where the Bob's chips are initially placed.

Output

Print n integers. The i-th of them should be equal to the number of turns the game will last if Alice initially places her chip in the vertex i. If one of Bob's chips is already placed in vertex i, then the answer for vertex i is 0.

Examples

Input


5
2 4
3 1
3 4
3 5
2
4 5


Output


2 1 2 0 0


Input


8
4 1
8 4
4 5
6 4
2 5
4 3
1 7
3
2 8 3


Output


3 0 0 3 1 2 3 0


Input


10
2 5
4 3
7 3
7 2
5 8
3 6
8 10
7 9
7 1
4
10 6 9 1


Output


0 2 2 2 2 0 2 2 0 0

Solution (in CPP):

Solution:
#include <algorithm>
#include <array>
#include <cassert>
#include <chrono>
#include <cmath>
#include <cstring>
#include <functional>
#include <iomanip>
#include <iostream>
#include <map>
#include <numeric>
#include <queue>
#include <random>
#include <set>
#include <vector>
using namespace std;

template<typename A, typename B> ostream& operator<<(ostream &os, const pair<A, B> &p) { return os << '(' << p.first << ", " << p.second << ')'; }
template<typename T_container, typename T = typename enable_if<!is_same<T_container, string>::value, typename T_container::value_type>::type> ostream& operator<<(ostream &os, const T_container &v) { os << '{'; string sep; for (const T &x : v) os << sep << x, sep = ", "; return os << '}'; }

void dbg_out() { cerr << endl; }
template<typename Head, typename... Tail> void dbg_out(Head H, Tail... T) { cerr << ' ' << H; dbg_out(T...); }

#ifdef NEAL_DEBUG
#define dbg(...) cerr << "(" << #__VA_ARGS__ << "):", dbg_out(__VA_ARGS__)
#else
#define dbg(...)
#endif

const int INF = int(1e9) + 5;

struct BFS {
    int n;
    vector<vector<int>> adj;
    vector<int> dist;
    vector<int> parent;

    BFS(int _n = 0) {
        init(_n);
    }

    void init(int _n) {
        n = _n;
        adj.assign(n, {});
    }

    void add_directional_edge(int a, int b) {
        adj[a].push_back(b);
    }

    void add_bidirectional_edge(int a, int b) {
        adj[a].push_back(b);
        adj[b].push_back(a);
    }

    void bfs_check(queue<int> &q, int node, int from, int new_dist) {
        if (new_dist < dist[node]) {
            dist[node] = new_dist;
            parent[node] = from;
            q.push(node);
        }
    }

    void bfs(const vector<int> &source) {
        if (n == 0) return;

        queue<int> q;
        dist.assign(n, INF);
        parent.assign(n, -1);

        for (int src : source)
            bfs_check(q, src, -1, 0);

        while (!q.empty()) {
            int top = q.front(); q.pop();

            for (int neigh : adj[top])
                bfs_check(q, neigh, top, dist[top] + 1);
        }
    }
};

BFS bfs;
vector<int> answers;

struct edge {
    int node;

    edge(int _node = -1) : node(_node) {}
};

struct centroid_decomposition {
    int N;
    vector<vector<edge>> adj;
    vector<int> depth;
    vector<int> subtree_size;
    vector<int> centroid_parent;
    vector<int> nodes;
    vector<int> depth_max;

    centroid_decomposition(int _N = 0) {
        init(_N);
    }

    void init(int _N) {
        N = _N;
        adj.assign(N, {});
        depth.resize(N);
        subtree_size.resize(N);
        centroid_parent.assign(N, -1);
        depth_max.assign(N + 1, 0);
    }

    void add_edge(int u, int v) {
        assert(u != v);
        adj[u].emplace_back(v);
        adj[v].emplace_back(u);
    }

    void erase_edge(int from, int to) {
        for (edge &e : adj[from])
            if (e.node == to) {
                swap(e, adj[from].back());
                adj[from].pop_back();
                return;
            }

        assert(false);
    }

    int dfs(int node, int parent = -1, int subroot = -1) {
        if (parent < 0) {
            subroot = node;
            nodes.clear();
        }

        depth[node] = parent < 0 ? 0 : depth[parent] + 1;
        subtree_size[node] = 1;
        nodes.push_back(node);

        for (edge &e : adj[node])
            if (e.node != parent)
                subtree_size[node] += dfs(e.node, node, parent < 0 ? e.node : subroot);

        return subtree_size[node];
    }

    int centroid(int root) {
        int n = dfs(root);
        bool found;

        // Repeatedly move to the subtree that is at least half of the tree, if such a subtree exists.
        do {
            found = false;

            for (edge &e : adj[root])
                if (subtree_size[e.node] < subtree_size[root] && 2 * subtree_size[e.node] >= n) {
                    root = e.node;
                    found = true;
                    break;
                }
        } while (found);

        return root;
    }

    void solve(int root) {
        root = centroid(root);

        for (int node : nodes)
            if (node != root)
                centroid_parent[node] = root;

        dfs(root);
        int n = int(nodes.size());

        for (int node : nodes) {
            int reach = min(bfs.dist[node] - 1 - depth[node], n);

            if (reach >= 0)
                depth_max[reach] = max(depth_max[reach], bfs.dist[node]);
        }

        for (int i = n - 1; i >= 0; i--)
            depth_max[i] = max(depth_max[i], depth_max[i + 1]);

        for (int node : nodes)
            answers[node] = max(answers[node], depth_max[depth[node]]);

        for (int i = 0; i <= n; i++)
            depth_max[i] = 0;

        for (edge &e : adj[root]) {
            erase_edge(e.node, root);
            solve(e.node);
        }
    }
};

template<typename T_vector>
void output_vector(const T_vector &v, bool add_one = false, int start = -1, int end = -1) {
    if (start < 0) start = 0;
    if (end < 0) end = int(v.size());

    for (int i = start; i < end; i++)
        cout << v[i] + (add_one ? 1 : 0) << (i < end - 1 ? ' ' : '\n');
}


int main() {
    ios::sync_with_stdio(false);
#ifndef NEAL_DEBUG
    cin.tie(nullptr);
#endif

    int N;
    cin >> N;
    bfs.init(N);
    answers.assign(N, 0);
    centroid_decomposition CD(N);

    for (int e = 0; e < N - 1; e++) {
        int u, v;
        cin >> u >> v;
        u--; v--;
        bfs.add_bidirectional_edge(u, v);
        CD.add_edge(u, v);
    }

    int K;
    cin >> K;
    vector<int> bob(K);

    for (auto &b : bob) {
        cin >> b;
        b--;
    }

    bfs.bfs(bob);
    CD.solve(0);
    output_vector(answers);
}