**Note: The time limit for this problem is 5s, 2.5 times the default. The memory limit for this problem is 512MB, twice the default.**

Bessie has recently taken an interest in magic and needs to collect mana for a very important spell. Bessie has $N$ ($1\le N\le 18$) mana pools, the $i$th of which accumulates $m_i$ mana per second ($1\le m_i\le 10^8$). The pools are linked by a collection of $M$ ($0\le M\le N(N-1)$) directed edges $(a_i,b_i,t_i)$, meaning that she can travel from $a_i$ to $b_i$ in $t_i$ seconds ($1\le a_i, b_i\le N$, $a_i\neq b_i$, $1\le t_i\le 10^9$). Whenever Bessie is present at a pool, she can collect all the mana stored at that location, emptying it. At time $0$, all mana pools are empty, and Bessie can select any pool to start at.

Answer $Q$ ($1\le Q\le 2\cdot 10^5$) queries, each specified by two integers $s$ and $e$ ($1\le s\le 10^9$, $1\le e\le N$). For each query, determine the maximum amount of mana Bessie can collect in $s$ seconds if she must be at mana pool $e$ at the end of the $s$th second.

SAMPLE INPUT:

2 1
1 10
1 2 10
4
5 1
5 2
100 1
100 2

SAMPLE OUTPUT:

5
50
100
1090

First query: Bessie takes 5 mana from pool 1 after 5 seconds.

Second query: Bessie takes 50 mana from pool 2 after 5 seconds.

Third query: Bessie takes 100 mana from pool 1 after 100 seconds.

Fourth query: Bessie takes 90 mana from pool 1 after 90 seconds and 1000 mana from pool 2 after 100 seconds.

SAMPLE INPUT:

4 8
50000000 100000000 20000000 70000000
1 2 20
2 1 50
2 3 90
1 3 40
3 1 10
4 1 25
1 4 5
4 3 70
3
8 3
1000000000 1
500000 4

SAMPLE OUTPUT:

160000000
239999988050000000
119992550000000

An example where Bessie is able to collect much larger amounts of mana.

Problem credits: Benjamin Qi