101375: [AtCoder]ABC137 F - Polynomial Construction

Problem Statement

Given are a prime number $p$ and a sequence of $p$ integers $a_0, \ldots, a_{p-1}$ consisting of zeros and ones.

Find a polynomial of degree at most $p-1$, $f(x) = b_{p-1} x^{p-1} + b_{p-2} x^{p-2} + \ldots + b_0$, satisfying the following conditions:

• For each $i$ $(0 \leq i \leq p-1)$, $b_i$ is an integer such that $0 \leq b_i \leq p-1$.
• For each $i$ $(0 \leq i \leq p-1)$, $f(i) \equiv a_i \pmod p$.

Constraints

• $2 \leq p \leq 2999$
• $p$ is a prime number.
• $0 \leq a_i \leq 1$

Input

Input is given from Standard Input in the following format:

$p$
$a_0$ $a_1$ $\ldots$ $a_{p-1}$

Output

Print $b_0, b_1, \ldots, b_{p-1}$ of a polynomial $f(x)$ satisfying the conditions, in this order, with spaces in between.

It can be proved that a solution always exists. If multiple solutions exist, any of them will be accepted.

Sample Input 1

2
1 0

Sample Output 1

1 1

$f(x) = x + 1$ satisfies the conditions, as follows:

• $f(0) = 0 + 1 = 1 \equiv 1 \pmod 2$
• $f(1) = 1 + 1 = 2 \equiv 0 \pmod 2$

Sample Input 2

3
0 0 0

Sample Output 2

0 0 0

$f(x) = 0$ is also valid.

Sample Input 3

5
0 1 0 1 0

Sample Output 3

0 2 0 1 3