Description:
For an array $$$u_1, u_2, \ldots, u_n$$$, define
- a prefix maximum as an index $$$i$$$ such that $$$u_i>u_j$$$ for all $$$j<i$$$;
- a suffix maximum as an index $$$i$$$ such that $$$u_i>u_j$$$ for all $$$j>i$$$;
- an ascent as an index $$$i$$$ ($$$i>1$$$) such that $$$u_i>u_{i-1}$$$.
You are given three cost arrays: $$$[a_1, a_2, \ldots, a_n]$$$, $$$[b_1, b_2, \ldots, b_n]$$$, and $$$[c_0, c_1, \ldots, c_{n-1}]$$$.
Define the cost of an array that has $$$x$$$ prefix maximums, $$$y$$$ suffix maximums, and $$$z$$$ ascents as $$$a_x\cdot b_y\cdot c_z$$$.
Let the sum of costs of all permutations of $$$1,2,\ldots,n$$$ be $$$f(n)$$$. Find $$$f(1)$$$, $$$f(2)$$$, ..., $$$f(n)$$$ modulo $$$998\,244\,353$$$.
Input Format:
The first line contains an integer $$$n$$$ ($$$1\le n\le 700$$$).
The second line contains $$$n$$$ integers $$$a_1,\ldots,a_n$$$ ($$$0\le a_i<998\,244\,353$$$).
The third line contains $$$n$$$ integers $$$b_1,\ldots,b_n$$$ ($$$0\le b_i<998\,244\,353$$$).
The fourth line contains $$$n$$$ integers $$$c_0,\ldots,c_{n-1}$$$ ($$$0\le c_i<998\,244\,353$$$).
Output Format:
Print $$$n$$$ integers: the $$$i$$$-th one is $$$f(i)$$$ modulo $$$998\,244\,353$$$.
Note:
In the second example:
- Consider permutation $$$[1,2,3]$$$. Indices $$$1,2,3$$$ are prefix maximums. Index $$$3$$$ is the only suffix maximum. Indices $$$2,3$$$ are ascents. In conclusion, it has $$$3$$$ prefix maximums, $$$1$$$ suffix maximums, and $$$2$$$ ascents. Therefore, its cost is $$$a_3b_1c_2=12$$$.
- Permutation $$$[1,3,2]$$$ has $$$2$$$ prefix maximums, $$$2$$$ suffix maximums, and $$$1$$$ ascent. Its cost is $$$6$$$.
- Permutation $$$[2,1,3]$$$ has $$$2$$$ prefix maximums, $$$1$$$ suffix maximum, and $$$1$$$ ascent. Its cost is $$$4$$$.
- Permutation $$$[2,3,1]$$$ has $$$2$$$ prefix maximums, $$$2$$$ suffix maximums, and $$$1$$$ ascent. Its cost is $$$6$$$.
- Permutation $$$[3,1,2]$$$ has $$$1$$$ prefix maximum, $$$2$$$ suffix maximums, and $$$1$$$ ascent. Its cost is $$$3$$$.
- Permutation $$$[3,2,1]$$$ has $$$1$$$ prefix maximum, $$$3$$$ suffix maximums, and $$$0$$$ ascents. Its cost is $$$3$$$.
The sum of all permutations' costs is $$$34$$$, so $$$f(3)=34$$$.