Bold Play to a Quadrupling Goal

You have $\$2$ and want to reach $\$8$ on an even-money game you win with probability $p=0.4$. You use bold play: each round you stake the most that keeps you from overshooting the goal, i.e. $\min(\text{current wealth},\ 8-\text{current wealth})$. What is the probability bold play reaches the $\$8$ goal?