E. Rectangular Congruence
time limit per test
1 second
memory limit per test
256 megabytes
input
standard input
output
standard output

You are given a prime number $$$n$$$, and an array of $$$n$$$ integers $$$b_1,b_2,\ldots, b_n$$$, where $$$0 \leq b_i < n$$$ for each $$$1 \le i \leq n$$$.

You have to find a matrix $$$a$$$ of size $$$n \times n$$$ such that all of the following requirements hold:

  • $$$0 \le a_{i,j} < n$$$ for all $$$1 \le i, j \le n$$$.

  • $$$a_{r_1, c_1} + a_{r_2, c_2} \not\equiv a_{r_1, c_2} + a_{r_2, c_1} \pmod n$$$ for all positive integers $$$r_1$$$, $$$r_2$$$, $$$c_1$$$, and $$$c_2$$$ such that $$$1 \le r_1 < r_2 \le n$$$ and $$$1 \le c_1 < c_2 \le n$$$.
  • $$$a_{i,i} = b_i$$$ for all $$$1 \le i \leq n$$$.

Here $$$x \not \equiv y \pmod m$$$ denotes that $$$x$$$ and $$$y$$$ give different remainders when divided by $$$m$$$.

If there are multiple solutions, output any. It can be shown that such a matrix always exists under the given constraints.

Input

The first line contains a single positive integer $$$n$$$ ($$$2 \le n < 350$$$).

The second line contains $$$n$$$ integers $$$b_1, b_2, \ldots, b_n$$$ ($$$0 \le b_i < n$$$) — the required values on the main diagonal of the matrix.

It is guaranteed that $$$n$$$ is prime.

Output

Print $$$n$$$ lines. On the $$$i$$$-th line, print $$$n$$$ integers $$$a_{i, 1}, a_{i, 2}, \ldots, a_{i, n}$$$, each separated with a space.

If there are multiple solutions, output any.

Examples
Input
2
0 0
Output
0 1 
0 0
Input
3
1 1 1
Output
1 2 2
1 1 0
1 0 1
Input
5
1 4 1 2 4
Output
1 0 1 3 4
1 4 3 1 0
2 4 1 0 2
1 2 2 2 2
2 2 0 1 4
Note

In the first example, the answer is valid because all entries are non-negative integers less than $$$n = 2$$$, and $$$a_{1,1}+a_{2,2} \not\equiv a_{1,2}+a_{2,1} \pmod 2$$$ (because $$$a_{1,1}+a_{2,2} = 0 + 0 \equiv 0 \pmod 2$$$ and $$$a_{1,2}+a_{2,1} = 1 + 0 \equiv 1 \pmod 2 $$$). Moreover, the values on the main diagonals are equal to $$$0,0$$$ as required.

In the second example, the answer is correct because all entries are non-negative integers less than $$$n = 3$$$, and the second condition is satisfied for all quadruplets $$$(r_1, r_2, c_1, c_2)$$$. For example:

  • When $$$r_1=1$$$, $$$r_2=2$$$, $$$c_1=1$$$ and $$$c_2=2$$$, $$$a_{1,1}+a_{2,2} \not\equiv a_{1,2}+a_{2,1} \pmod 3$$$ because $$$a_{1,1}+a_{2,2} = 1 + 1 \equiv 2 \pmod 3$$$ and $$$a_{1,2}+a_{2,1} = 2 + 1 \equiv 0 \pmod 3 $$$.
  • When $$$r_1=2$$$, $$$r_2=3$$$, $$$c_1=1$$$, and $$$c_2=3$$$, $$$a_{2,1}+a_{3,3} \not\equiv a_{2,3}+a_{3,1} \pmod 3$$$ because $$$a_{2,1}+a_{3,3} = 1 + 1 \equiv 2 \pmod 3$$$ and $$$a_{2,3}+a_{3,1} = 0 + 1 \equiv 1 \pmod 3 $$$.
Moreover, the values on the main diagonal are equal to $$$1,1,1$$$ as required.