
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 $\triangle ABC$, $AB=6$, $BC=10$, and $CA=8$, with $\overline{AH}$ as an altitude. Points $D$ and $E$ are 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. Additionally, $D$ and $E$ are positioned so that $BD = DE = EC$. Determine the length of segment $PH$.

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