G. Sum of Prefix Sumstime limit per test6 secondsmemory limit per test512 megabytesinputstandard inputoutputstandard outputWe define the sum of prefix sums of an array [s_1, s_2, \dots, s_k] as s_1 + (s_1 + s_2) + (s_1 + s_2 + s_3) + \dots + (s_1 + s_2 + \dots + s_k).You are given a tree consisting of n vertices. Each vertex i has an integer a_i written on it. We define the value of the simple path from vertex u to vertex v as follows: consider all vertices appearing on the path from u to v, write down all the numbers written on these vertices in the order they appear on the path, and compute the sum of prefix sums of the resulting sequence.Your task is to calculate the maximum value over all paths in the tree.InputThe first line contains one integer n (2 \le n \le 150000) — the number of vertices in the tree.Then n - 1 lines follow, representing the edges of the tree. Each line contains two integers u_i and v_i (1 \le u_i, v_i \le n, u_i \ne v_i), denoting an edge between vertices u_i and v_i. It is guaranteed that these edges form a tree.The last line contains n integers a_1, a_2, ..., a_n (1 \le a_i \le 10^6).OutputPrint one integer — the maximum value over all paths in the tree.ExampleInput
4
4 2
3 2
4 1
1 3 3 7
Output
36
NoteThe best path in the first example is from vertex 3 to vertex 1. It gives the sequence [3, 3, 7, 1], and the sum of prefix sums is 36.
4
4 2
3 2
4 1
1 3 3 7

36 
['data structures', 'divide and conquer', 'geometry', 'trees', '*2700']