
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
A right triangle $ABC$ has a right angle at $C$. A circle is constructed on its leg $BC$, which has a length of 26, as its diameter. From point $A$, a tangent $AP$ is drawn to this circle, different from $AC$. A perpendicular $PH$ is dropped to the segment $BC$ and intersects segment $AB$ at point $Q$. Find the area of triangle $BPQ$, given that $BH : CH = 4 : 9$.

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