Problem:
Score : 300 points

Problem Statement

Given are positive integers N, M, Q, and Q quadruples of integers ( a_i , b_i , c_i , d_i ).

Consider a sequence A satisfying the following conditions:

A is a sequence of N positive integers.

1 \leq A_1 \leq A_2 \le \cdots \leq A_N \leq M.

Let us define a score of this sequence as follows:

The score is the sum of d_i over all indices i such that A_{b_i} - A_{a_i} = c_i. (If there is no such i, the score is 0.)

Find the maximum possible score of A.

Constraints

All values in input are integers.

2 ≤ N ≤ 10

1 \leq M \leq 10

1 \leq Q \leq 50

1 \leq a_i < b_i \leq N ( i = 1, 2, ..., Q )

0 \leq c_i \leq M - 1 ( i = 1, 2, ..., Q )

(a_i, b_i, c_i) \neq (a_j, b_j, c_j) (where i \neq j)

1 \leq d_i \leq 10^5 ( i = 1, 2, ..., Q )

Input

Input is given from Standard Input in the following format:

N M Q
a_1 b_1 c_1 d_1
:
a_Q b_Q c_Q d_Q

Output

Print the maximum possible score of A.

Sample Input 1

3 4 3
1 3 3 100
1 2 2 10
2 3 2 10

Sample Output 1

110

When A = \{1, 3, 4\}, its score is 110. Under these conditions, no sequence has a score greater than 110, so the answer is 110.

Sample Input 2

4 6 10
2 4 1 86568
1 4 0 90629
2 3 0 90310
3 4 1 29211
3 4 3 78537
3 4 2 8580
1 2 1 96263
1 4 2 2156
1 2 0 94325
1 4 3 94328

Sample Output 2

357500

Sample Input 3

10 10 1
1 10 9 1

Sample Output 3

1

Solution in Python (3.8.2) (judge verdict: Time Limit Exceeded)

import numpy as np
from itertools import product


def feasible(abcd, N, M):
    # assume abcd sorted
    A = [0] * (N+1)
    a, b, c, d = abcd.pop(0)
    A[a] = 1
    A[b] = c + 1
    while len(abcd) > 0:
        a, b, c, d = abcd.pop(0)
        if A[a] == 0 and A[b] == 0:
            abcd.append([a, b, c, d])
        elif A[a] != 0 and A[b] != 0:
            if A[b] - A[a] != c:
                return False
        elif A[b] == 0:
            A[b] = A[a] + c
            if A[b] > M:
                return False
            if max(A[:b]) > A[b]:
                return False
            if min([x for x in A[b:] if x > 0]) < A[b]:
                return False
        else:  # A[a] == 0
            A[a] = A[b] - c
            if A[a] < 1:
                return False
            if max(A[:a]) > A[a]:
                return False
            if min([x for x in A[a:] if x > 0]) < A[a]:
                return False
    return True


def main():
    N, M, Q = [int(x) for x in input().split()]
    abcd = [[int(x) for x in input().split()] for _ in range(Q)]
    if Q == 1:
        return abcd[0][3]
    abcd.sort()
    sf = [[sum([x[3] if y else 0 for x,y in zip(abcd, f)]), f]
          for f in product([1,0], repeat=Q) if f.count(True)>=2]
    sf.sort(reverse=True)
    for s, f in sf:
        target = [x for x,y in zip(abcd, f) if y]
        if feasible(target, N, M):
            return s
    return max([x[3] for x in abcd])


print(main())