You are given a positive integer $n > 1$. What is the largest possible number of integers that can be chosen from the set $\{1, 2, 3, \ldots, 2^n\}$ so that for any two different chosen integers $a, b$, the value $a^k + b^k$ is not divisible by $2^n$ for any positive integer $k$? Proposed by Oleksii Masalitin