Switches

Consider a game involving four switches, each represented by a non-negative integer indicating its position. The first switch can be set between $0$ and $a$, the second between $0$ and $b$, the third between $0$ and $c$, and the fourth between $0$ and $d$.

Initially, all switches are at position $0$. The objective is to move each switch to its maximum position—$a$, $b$, $c$, and $d$ respectively—in the fewest moves possible.

During each move, you can choose one switch that is not yet at its maximum position and increase its value by $1$. Simultaneously, each of the other switches may independently change its position: with equal probability, a switch can increase by $1$, decrease by $1$, or stay the same. If a switch is at position $0$, it can only increase by $1$ or remain unchanged. Similarly, if a switch is at its maximum position, it can only decrease by $1$ or remain unchanged, each with equal probability.

Your task is to calculate the average number of moves needed to bring all switches from position ($0$, $0$, $0$, $0$) to ($a$, $b$, $c$, $d$), assuming optimal play.