Consider a rectangle $ABCD$ with $AB = 30$ and $BC = 60$. Construct a circle $T$ whose diameter is $AD$. Construct another circle $S$ whose diameter is $AB$. Let circles $S$ and $T$ intersect at a point $P$ such that $P \neq A$. Let the line $AP$ intersect $BC$ at point $E$. Define point $F$ on $AB$ such that the line $EF$ is tangent to the circle with diameter $AD$. Find the area of triangle $\triangle AEF$.

Let's think step by step and output the final answer within \boxed{}.
user