
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
In triangle \( \triangle ABC \), side \( AB = 13 \), side \( BC = 14 \), and side \( CA = 15 \). An altitude \( \overline{AH} \) is drawn from \( A \) to \( BC \). Points \( D \) and \( E \) are situated on sides \( \overline{AC} \) and \( \overline{AB} \) respectively, such that \( \overline{BD} \) and \( \overline{CE} \) are angle bisectors, intersecting \( \overline{AH} \) at points \( Q \) and \( P \) respectively. Determine the length of segment \( PQ \).

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