
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
The incircle of a scalene triangle  $ABC$  touches the sides  $BC, CA$ , and  $AB$  at points  $D, E$ , and  $F$ , respectively. Triangles  $APE$  and  $AQF$  are constructed outside the triangle so that \[AP =PE, AQ=QF, \angle APE=\angle ACB,\text{ and }\angle AQF =\angle ABC.\]Let  $M$  be the midpoint of  $BC$ . Find  $\angle QMP$  in terms of the angles of the triangle  $ABC$ .

*Iran, Shayan Talaei*


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