PDA that accepts if there is an "a" in the second half of the string How could I build a PDA on $Σ = \{a,b\}$ that accepts if there is an "a" in the second half of the string?

Formally the language is the following: $L = \{xy \mid x ∈ Σ^*, y ∈ Σ^*aΣ^* \land |x| ≥ |y|\}$.
The PDA has three phases. In the first phase, it adds $A$ to the stack per character. It switches to the second phase nondeterministically. In the second phase, it removes $A$ per character. It switches to the third phase nondeterministically, during which it empties the stack. The PDA accepts if it encountered an $a$ during the second phase.