
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
Triangle \( ABC \) has side lengths \( AB = 14 \), \( AC = 13 \), and \( BC = 15 \). Point \( D \) is chosen in the interior of \( \overline{AB} \), and point \( E \) is selected uniformly at random from \( \overline{AD} \). Point \( F \) is then defined to be the intersection point of the perpendicular to \( \overline{AB} \) at \( E \) and the union of segments \( \overline{AC} \) and \( \overline{BC} \). Suppose that \( D \) is chosen such that the expected value of the length of \( \overline{EF} \) is maximized. Find \( AD \).

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