
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 right triangle $ABC$, hypotenuse $BC$ has a length of $m$. A circle with diameter $n$ $ (n < \frac{m}{2})$ is drawn with the midpoint of $BC$, $O$, as its center. The circle intersects $BC$ at points $P$ and $Q$. Find the value of $|AP|^2 + |AQ|^2 + |PQ|^2$.

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