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≤N≤18) mana pools, the ith of which accumulates mi mana per second (1≤mi≤108). The pools are linked by a collection of M (0≤M≤N(N−1)) directed edges (ai,bi,ti), meaning that she can travel from ai to bi in ti seconds (1≤ai,bi≤N, ai≠bi, 1≤ti≤109). 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≤Q≤2⋅105) queries, each specified by two integers s and e (1≤s≤109, 1≤e≤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 sth second.
INPUT FORMAT (input arrives from the terminal / stdin):
First line contains N and M.
Next line contains m1,m2,…,mN.
Next M lines contain ai,bi,ti. No ordered pair (ai,bi) 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≤10,Q≤100
Inputs 5-9: N≤10
Inputs 10-14: Q≤100
Inputs 15-17: N=16
Inputs 18-20: N=17
Inputs 21-24: No additional constraints.
Problem credits: Benjamin Qi