Plug Walk

The plug performs a random walk on the set $S = \{0,1,\dots, m\}$, where $m > 0$. The plug starts at position $0 \leq k < m$. Each step, the plug moves up or down $1$ step with equal probability. The plug has opps at $0$, so they move back upwards with probability $1$ whenever they hit $0$. Find the expected amount of steps that occurs until the plug first reaches position $m$ when $k = 4$ and $m = 20$.