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Ex - Construct a Matrix

Score : $600$ points

Problem Statement

Determine whether there is an $N$-by-$N$ matrix $X$ that satisfies the following conditions, and present one such matrix if it exists. (Let $x_{i,j}$ denote the element of $X$ at the $i$-th row from the top and $j$-th column from the left.)

• $x_{i,j} \in \{ 0,1,2 \}$ for every $i$ and $j$ $(1 \leq i,j \leq N)$.
• The following holds for each $i=1,2,\ldots,Q$.
  • Let $P = \prod_{a_i \leq j \leq b_i} \prod_{c_i \leq k \leq d_i} x_{j,k}$. Then, $P$ is congruent to $e_i$ modulo $3$.

Constraints

• $1 \leq N,Q \leq 2000$
• $1 \leq a_i \leq b_i \leq N$
• $1 \leq c_i \leq d_i \leq N$
• $e_i \in \{0,1,2 \}$
• All values in the input are integers.

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Input

The input is given from Standard Input in the following format:

$N$ $Q$

$a_1$ $b_1$ $c_1$ $d_1$ $e_1$

$\vdots$

$a_Q$ $b_Q$ $c_Q$ $d_Q$ $e_Q$

Output

If no matrix $X$ satisfies the conditions, print No.

If there is a matrix $X$ that satisfies them, print Yes in the first line and one such instance of $X$ in the following format in the second and subsequent lines.

``` $x_{1,1}$ $x_{1,2}$ $\ldots$ $x_{1,N}$ $x_{2,1}$ $x_{2,2}$ $\ldots$ $x_{2,N}$ $\vdots$ $x_{N,1}$ $x_{N,2}$ $\ldots$ $x_{N,N}$ ```

If multiple matrices satisfy the conditions, any of them will be accepted.

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2 3
1 1 1 2 0
1 2 2 2 1
2 2 1 2 2

Yes
0 2
1 2

For example, for $i=2$, we have $P = \prod_{a_2 \leq j \leq b_2} \prod_{c_2 \leq k \leq d_2} x_{j,k}= \prod_{1 \leq j \leq 2} \prod_{2 \leq k \leq 2} x_{j,k}=x_{1,2} \times x_{2,2}$.
In this sample output, we have $x_{1,2}=2$ and $x_{2,2}=2$, so $P=2 \times 2 = 4$, which is congruent to $e_2=1$ modulo $3$.
It can be similarly verified that the condition is also satisfied for $i=1$ and $i=3$.

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4 4
1 4 1 4 0
1 4 1 4 1
1 4 1 4 2
1 4 1 4 0

No