
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Let \(ABC\) 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 \(ABD\) and \(ACD\), and let the common external tangent of \(\omega_1\) and \(\omega_2\) (other than \(BC\)) intersect \(AD\) at \(E\). Compute the length of \(AE\).

Present the answer in LaTex format: \boxed{Your answer}
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