KöMaL 2025 B5492

Kornél thinks about a closed subinterval of $I=[0, 2^k]$ (where $k$ is a positive integer) with integer endpoints and length at least $1$. Kristóf can ask the following question: he can choose an arbitrary closed subinterval with integer length, but not necessarily integer endpoints, and Kornél tells him the length of the intersection of the interval he picked and the interval chosen by Kristóf. (The answer is $0$ if the intersection of the two intervals is empty or consists of a single point.) Find the smallest number of questions with which Kristóf can guess the interval chosen by Kornél in all cases.