 ## Problem 2. Mana Collection

Contest has ended. **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.

#### INPUT FORMAT (input arrives from the terminal / stdin):

First line contains $N$ and $M$.

Next line contains $m_1,m_2,\dots, m_N$.

Next $M$ lines contain $a_i,b_i,t_i$. No ordered pair $(a_i,b_i)$ appears more than once in the input.

Next line contains $Q$.

Next $Q$ lines contain two integers $s$ and $e$.

#### OUTPUT FORMAT (print output to the terminal / stdout):

$Q$ lines, one for each query.

#### 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.

#### SCORING:

• Inputs 3-4: $N\le 10, Q\le 100$
• Inputs 5-9: $N\le 10$
• Inputs 10-14: $Q\le 100$
• Inputs 15-17: $N = 16$
• Inputs 18-20: $N = 17$
• Inputs 21-24: No additional constraints.

Problem credits: Benjamin Qi

Contest has ended. No further submissions allowed.