Consider a special binary counter represented as n bits: bn$1$ bn$2...$ b$1$ b$0.$ For this special counter, the cost of flipping the ith bit is $2$i tokens. For example, b$0$ costs $1$ token to flip, b$1$ costs $2$ tokens to flip, b$2$ costs $4$ tokens to flip, etc. We wish to analyze the cost of performing k $=2$n increments of this n$-$bit counter. $($Note that n is not a constant.$)$