MEX and XOR 2

You are given an integer $N$. Consider the sequence $A = (0, 1, 2, \ldots, N)$. A non-empty subsequence$^{\dagger}$ $B$ is called *good* if $\text{MEX}(B) = \text{XOR}(B)$ $^{\ddagger}$. Count the number of good non-empty subsequences of $A$. Since the answer may be large, print it modulo $998244353$. --- $^{\dagger}$ A sequence $B$ is said to be a subsequence of $A$ if it can be formed by