Dirichlet $k$-th root

Mathematician Pang learned Dirichlet convolution during the previous camp. However, compared with deep reinforcement learning, it was 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 a 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 $1$. 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 any of them.