Do the previous exercise again, but this time prove the result using the well-ordering of Nat. [Hint. Use RAA; define
$$
U=|k| \text { there exists } \gamma \text { such that }
$$
( $\gamma \in A \& \gamma$ has unbalanced parentheses \& $k=\operatorname{rank}(\gamma)))$.
Assume that $U$ is nonempty, and apply WO to it.]