Keen Tree Calculation
Description
There is a tree of $N$ vertices and $N-1$ edges. The $i$-th edge connects vertices $U_i$ and $V_i$ and has a length of $W_i$.
Chaneka, the owner of the tree, asks you $Q$ times. For the $j$-th question, the following is the question format:
- $X_j$ $K_j$ – If each edge that contains vertex $X_j$ has its length multiplied by $K_j$, what is the diameter of the tree?
Notes:
- Each of Chaneka's question is independent, which means the changes in edge length do not influence the next questions.
- The diameter of a tree is the maximum possible distance between two different vertices in the tree.
The first line contains a single integer $N$ ($2\leq N\leq10^5$) — the number of vertices in the tree.
The $i$-th of the next $N-1$ lines contains three integers $U_i$, $V_i$, and $W_i$ ($1 \leq U_i,V_i \leq N$; $1\leq W_i\leq10^9$) — an edge that connects vertices $U_i$ and $V_i$ with a length of $W_i$. The edges form a tree.
The $(N+1)$-th line contains a single integer $Q$ ($1\leq Q\leq10^5$) — the number of questions.
The $j$-th of the next $Q$ lines contains two integers $X_j$ and $K_j$ as described ($1 \leq X_j \leq N$; $1 \leq K_j \leq 10^9$).
Output $Q$ lines with an integer in each line. The integer in the $j$-th line represents the diameter of the tree on the $j$-th question.
Input
The first line contains a single integer $N$ ($2\leq N\leq10^5$) — the number of vertices in the tree.
The $i$-th of the next $N-1$ lines contains three integers $U_i$, $V_i$, and $W_i$ ($1 \leq U_i,V_i \leq N$; $1\leq W_i\leq10^9$) — an edge that connects vertices $U_i$ and $V_i$ with a length of $W_i$. The edges form a tree.
The $(N+1)$-th line contains a single integer $Q$ ($1\leq Q\leq10^5$) — the number of questions.
The $j$-th of the next $Q$ lines contains two integers $X_j$ and $K_j$ as described ($1 \leq X_j \leq N$; $1 \leq K_j \leq 10^9$).
Output
Output $Q$ lines with an integer in each line. The integer in the $j$-th line represents the diameter of the tree on the $j$-th question.
7
5 1 2
1 4 2
3 4 1
2 5 3
6 1 6
4 7 2
2
4 3
3 2
3
1 2 1000000000
2 3 1000000000
1
2 1000000000
18
11
2000000000000000000
Note
In the first example, the following is the tree without any changes.
The following is the tree on the $1$-st question.
The maximum distance is between vertices $6$ and $7$, which is $6+6+6=18$, so the diameter is $18$.
The following is the tree on the $2$-nd question.
The maximum distance is between vertices $2$ and $6$, which is $3+2+6=11$, so the diameter is $11$.