Let $\triangle ABC$ be a triangle with $AB = 13$, $BC = 14$, and $CA = 15$. The altitude from $A$ intersects $BC$ at $D$.

Let $\omega_1$ and $\omega_2$ be the incircles of $\triangle ABD$ and $\triangle ACD$, respectively. The common external tangent of $\omega_1$ and $\omega_2$ (other than $BC$) intersects $AD$ at $E$. Compute the length of $AE$.

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