In a pushdown automaton \( P = (Q, \Sigma, \Gamma, \delta, q_0, F) \), a transition of the form
where \( p, q \in Q \), \( a \in \Sigma \cup \{\epsilon\} \), and \( X, Y \in \Gamma \cup \{\epsilon\} \), represents \[ (q, Y) \in \delta(p, a, X). \] Consider the following pushdown automaton over the input alphabet \( \Sigma = \{a, b\} \) and stack alphabet \( \Gamma = \{\#, A\} \):
The number of strings of length 100 accepted by the above pushdown automaton is \(\underline{\hspace{2cm}}\).