Ex - Construct a Matrix

AtCoder

IDabc276_h

Time4000ms

Memory256MB

Difficulty—

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. ## Input The input is given from Standard Input in the following format: $N$ $Q$ $a_1$ $b_1$ $c_1$ $d_1$ $e_1$ $\vdots$ $a_Q$ $b_Q$ $c_Q$ $d_Q$ $e_Q$ [samples]

Samples

Input #1

Output #1

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 $i=3$.

Input #2

Output #2

No

API Response (JSON)

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      "content": "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...",
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