Given a binary array $A$ (consisting of 1's and 0's) of length $n\ (=438276543893501974)$, we define a time-dependent transformation: $$A[i]^{(t)} = A[i+1]^{(t-1)}\ \oplus\ A[i+2]^{(t-1)} \oplus\ ...\ \oplus\ A[n]^{(t-1)}$$

where $\oplus$ denotes the bitwise XOR operation. At each time step, the last element of the array becomes 0, $A[n]^{(t)} = 0$ for all $t > 0$.

For a configuration of the array $A$: $$F(A) = \sum_{t=0}^{\infty} \sum_{i=1}^{n} A[i]^{(t)}$$

Your task is to compute the sum of $F(A)$ over all $2^n$ possible configurations of $A$. Since the answer may be large, report it modulo $998,244,353$.