Count numbers that can be generated by flipping bits according to position intervals
Hint: For given $N$ and a range $[L,H]$, let $v \in \mathbb{Z}_2^N$ be the characteristic vector of $[L,H]$, i.e. $v_i = 1$ if $L \leq i \leq H$ and $v_i = 0$ otherwise. Given a list of ranges $[L_j,H_j]$ and correspond vectors $v_j$, how can you represent the characteristic vectors of all numbers reachable by the operations you mention? What is the relevant parameter?

Another hint: What happens if all $L_i$ are distinct? How can you handle cases in which $L_i = L_j$ for some $i \neq j$? Try to reduce it to the former case.