23453번 - Dirichlet $k$-th root 스페셜 저지다국어

Mathematician Pang learned Dirichlet convolution during the previous camp. However, compared with deep reinforcement learning, it's too easy for him. Therefore, he did something special. 

If $f,g: \{1,2,\ldots,n\} \to \mathbb {Z} $ are two functions from the positive integers to the integers, the Dirichlet convolution $f * g$ is a new function defined by: $$(f * g)(n)  =\sum_{d \mid n}f(d)g ({\frac {n}{d}}) .$$

We define the $k$-th power of an function $g=f^k$ by $$ f^{k}=\underbrace {f * \dots * f} _{k {\textrm {times}}}.$$

In this problem, we want to solve the inverse problem: Given $g$ and $k$, you need to find a function $f$ such that $g=f^k$.

Moreover, there is an additional constraint that $f(1)$ and $g(1)$ must equal to $1$. And all the arithmetic operations are done on $\mathbb{F}_{p}$ where $p=998244353$, which means that in the Dirichlet convolution, $(f * g)(n)  =\left(\sum_{d \mid n}f(d)g ({\frac {n}{d}})\right) \bmod p$.

The first line contains two integers $n$ and $k (2\leq n\leq 10^5,1\leq k<998244353)$ .

The second line contains n integers $g(1), g(2),..., g(n)$ ($0\le g(i)<998244353, g(1)=1$).

If there is no solution, output $-1$.

Otherwise, output $f(1), f(2), ..., f(n)$ ($0\le f(i)<998244353, f(1)=1$). If there are multiple solutions, print anyone.