Given a rooted tree on $$$n$$$ vertices with values $$$a_1, \dots, a_n$$$ written on the vertices, we define the score of a vertex as the product of $$$a_u$$$ for all vertices $$$u$$$ in the subtree of $$$v$$$.

However, you've lost the tree and just the values remain. All you remember is the following,

• the tree had $$$n$$$ vertices labelled from $$$1$$$ to $$$n$$$,
• the tree was rooted at the vertex with label $$$1$$$,
• for every non-root vertex, its parent had a strictly smaller label.

Even though there are $$$(n - 1)!$$$ such trees possible, you try to compute the score of each vertex using just this information.

Output the expected score, modulo $$$998\,244\,353$$$, of each vertex on uniformly randomly sampling a tree satisfying the above constraints.

Formally, let $$$M = 998\,244\,353$$$. It can be shown that the expected score can be expressed as an irreducible fraction $$$\frac{p}{q}$$$, where $$$p$$$ and $$$q$$$ are integers and $$$q \not \equiv 0 \pmod{M}$$$. Output the integer equal to $$$p \cdot q^{-1} \bmod M$$$. In other words, output an integer $$$x$$$ such that $$$0 \le x \lt M$$$ and $$$x \cdot q \equiv p \pmod{M}$$$.