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C - Many Requirements

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 $A$.

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

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 $110$.

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

357500

10 10 1
1 10 9 1

1