
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
Consider a right -angle triangle  $ABC$  with  $A=90^{o}$ ,  $AB=c$  and  $AC=b$ . Let  $P\in AC$  and  $Q\in AB$  such that  $\angle APQ=\angle ABC$  and  $\angle AQP = \angle ACB$ . Calculate  $PQ+PE+QF$ , where  $E$  and  $F$  are the projections of  $B$  and  $Q$  onto  $BC$ , respectively.

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