A $2-$Stack Nondeterministic Pushdown Automata $(2-$PDA$)$ is a PDA that has $2$ stacks that are used at each transition. At each transition, the machine reads a symbol from the input, pops specified symbols from each of the two stacks, pushes specified symbols onto each of the two stacks, and continues per the transition function. The machine accepts if there is some computation on its input string that causes it to reach an accept state.

$($a$)(4$ points$)$ Give a formal definition of a $2-$PDA, and a formal definition of what it means for a $2-$PDA to accept an input string.