$(5$ pts$)$ Define a pushdown automaton $($PDA$)$ for the language $L=\{\begin{array}{ccc}\text{w}\text{i}\text{n}& \{0,& 1{\}}^{*}|w\end{array}$ is of the form $0^n{1}^n$ for some $\{$:$n0\}$ :

I. List the states, input alphabet, stack alphabet, start state, and transition functions of the PDA.

II$.$ Explain how the PDA processes the string $"0011".$

III. IIlustrate the PDA's operation for the string $"000111"$ step$-$by$-$step.