Bitsetbaba and Power Grid

Bitsetbaba Ashiq works in a power grid, which consists of $2^K$ power stations (numbered $0$ through $2^K-1$). Initially, there are no connections between the power stations. One day, Ashiq's boss gave him $M$ integers $x_1, x_2, \ldots, x_M$ and asked Ashiq to add connections between some power stations. For each two different power stations $u$ and $v$, Ashiq should connect them directly if t