Description: You are given a rooted tree consisting of $$$n$$$ vertices. The vertices are numbered from $$$1$$$ to $$$n$$$, and the root is the vertex $$$1$$$. You are also given a score array $$$s_1, s_2, \ldots, s_n$$$. A multiset of $$$k$$$ simple paths is called valid if the following two conditions are both true. - Each path starts from $$$1$$$. - Let $$$c_i$$$ be the number of paths covering vertex $$$i$$$. For each pair of vertices $$$(u,v)$$$ ($$$2\le u,v\le n$$$) that have the same parent, $$$|c_u-c_v|\le 1$$$ holds. It can be shown that it is always possible to find at least one valid multiset. Find the maximum value among all valid multisets. Input Format: Each test contains multiple test cases. The first line contains a single integer $$$t$$$ ($$$1 \leq t \leq 10^4$$$) — the number of test cases. The description of the test cases follows. The first line of each test case contains two space-separated integers $$$n$$$ ($$$2 \le n \le 2 \cdot 10^5$$$) and $$$k$$$ ($$$1 \le k \le 10^9$$$) — the size of the tree and the required number of paths. The second line contains $$$n - 1$$$ space-separated integers $$$p_2,p_3,\ldots,p_n$$$ ($$$1\le p_i\le n$$$), where $$$p_i$$$ is the parent of the $$$i$$$-th vertex. It is guaranteed that this value describe a valid tree with root $$$1$$$. The third line contains $$$n$$$ space-separated integers $$$s_1,s_2,\ldots,s_n$$$ ($$$0 \le s_i \le 10^4$$$) — the scores of the vertices. It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$2 \cdot 10 ^ 5$$$. Output Format: For each test case, print a single integer — the maximum value of a path multiset. Note: In the first test case, one of optimal solutions is four paths $$$1 \to 2 \to 3 \to 5$$$, $$$1 \to 2 \to 3 \to 5$$$, $$$1 \to 4$$$, $$$1 \to 4$$$, here $$$c=[4,2,2,2,2]$$$. The value equals to $$$4\cdot 6+ 2\cdot 2+2\cdot 1+2\cdot 5+2\cdot 7=54$$$. In the second test case, one of optimal solution is three paths $$$1 \to 2 \to 3 \to 5$$$, $$$1 \to 2 \to 3 \to 5$$$, $$$1 \to 4$$$, here $$$c=[3,2,2,1,2]$$$. The value equals to $$$3\cdot 6+ 2\cdot 6+2\cdot 1+1\cdot 4+2\cdot 10=56$$$.