Contest Duration: - (local time) (100 minutes) Back to Home
Ex - Construct a Matrix /

Time Limit: 4 sec / Memory Limit: 1024 MB

### 問題文

• すべての i,j(1 \leq i,j \leq N) に対し、x_{i,j} \in \{ 0,1,2 \}
• i=1,2,\ldots,Q それぞれに対し次の条件が成立する。
• P = \prod_{a_i \leq j \leq b_i} \prod_{c_i \leq k \leq d_i} x_{j,k} とする。この時、P3 で割った余りは e_i に等しい。

### 制約

• 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 \}
• 入力はすべて整数

### 入力

N Q
a_1 b_1 c_1 d_1 e_1
\vdots
a_Q b_Q c_Q d_Q e_Q


### 出力

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}


### 入力例 1

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


### 出力例 1

Yes
0 2
1 2


この出力例において x_{1,2}=2, x_{2,2}=2 なので P=2 \times 2 = 4 であり、これを 3 で割った余りは e_2=1 に等しいです。
i=1,3 に対しても同様に条件を満たすことを確認できます。

### 入力例 2

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


### 出力例 2

No


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.

### 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.

### Sample Input 1

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


### Sample 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.

### Sample Input 2

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


### Sample Output 2

No