
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
\( ABC \) is an acute triangle with incircle \( \omega \). \( \omega \) is tangent to sides \( \overline{BC}, \overline{CA} \), and \( \overline{AB} \) at \( D, E \), and \( F \) respectively. \( P \) is a point on the altitude from \( A \) such that \( \Gamma \), the circle with diameter \( \overline{AP} \), is tangent to \( \omega \). \( \Gamma \) intersects \( \overline{AC} \) and \( \overline{AB} \) at \( X \) and \( Y \) respectively. Given \( XY = 8 \), \( AE = 15 \), and that the radius of \( \Gamma \) is 5, compute \( BD \cdot DC \).

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