## USACO 2023 January Contest, Platinum

## Problem 2. Mana Collection

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**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