
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
Given an obtuse triangle $ABC$ with an obtuse angle at $C$. Points $P$ and $Q$ are marked on its sides $AB$ and $BC$ respectively such that $\angle ACP = CPQ = 90^\circ$. Find the length of the segment $PQ$, given that $AC = 25$, $CP = 20$, and $\angle APC = \angle A + \angle B$.

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