GCD of the Submasks

For a non-negative integer $X$, the function $G(X)$ is defined as the greatest common divisor of all the integers generated by all submasks of the binary representation of $X$, i.e. integers obtained by replacing some (possibly none or all) $1$-s in this binary representation by $0$-s. You are given an integer $N$. Find $S = \sum_{X=1}^N X^{G(X)}$ modulo $998,244,353$. ### Input - The first