Markers on a Tree (Hard)

**This is the hard version of the problem. Here you need to find the sum of $f(S)$ over all $2^N$ subsets $S$ modulo $998244353$.** For a tree $T$ consisting of $N$ labelled nodes, labelled $1$ through $N$; consider having $2$ markers, marker $A$ initially located at node $1$ and marker $B$ located at node $N$. In one move, you can move **exactly** one of the $2$ markers from it's current node