Different Arrays

You are given an array $$$a$$$ consisting of $$$n$$$ integers.

You have to perform the sequence of $$$n-2$$$ operations on this array:

• during the first operation, you either add $$$a_2$$$ to $$$a_1$$$ and subtract $$$a_2$$$ from $$$a_3$$$, or add $$$a_2$$$ to $$$a_3$$$ and subtract $$$a_2$$$ from $$$a_1$$$;
• during the second operation, you either add $$$a_3$$$ to $$$a_2$$$ and subtract $$$a_3$$$ from $$$a_4$$$, or add $$$a_3$$$ to $$$a_4$$$ and subtract $$$a_3$$$ from $$$a_2$$$;
• ...
• during the last operation, you either add $$$a_{n-1}$$$ to $$$a_{n-2}$$$ and subtract $$$a_{n-1}$$$ from $$$a_n$$$, or add $$$a_{n-1}$$$ to $$$a_n$$$ and subtract $$$a_{n-1}$$$ from $$$a_{n-2}$$$.

So, during the $$$i$$$-th operation, you add the value of $$$a_{i+1}$$$ to one of its neighbors, and subtract it from the other neighbor.

For example, if you have the array $$$[1, 2, 3, 4, 5]$$$, one of the possible sequences of operations is:

• subtract $$$2$$$ from $$$a_3$$$ and add it to $$$a_1$$$, so the array becomes $$$[3, 2, 1, 4, 5]$$$;
• subtract $$$1$$$ from $$$a_2$$$ and add it to $$$a_4$$$, so the array becomes $$$[3, 1, 1, 5, 5]$$$;
• subtract $$$5$$$ from $$$a_3$$$ and add it to $$$a_5$$$, so the array becomes $$$[3, 1, -4, 5, 10]$$$.

So, the resulting array is $$$[3, 1, -4, 5, 10]$$$.

An array is reachable if it can be obtained by performing the aforementioned sequence of operations on $$$a$$$. You have to calculate the number of reachable arrays, and print it modulo $$$998244353$$$.