
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
Let  $ABC$  be a triangle with  $\angle A = 120^\circ$ ,  $I$  be the incenter, and  $M$  be the midpoint of  $BC$ . The line passing through  $M$  and parallel to  $AI$  meets the circle with diameter  $BC$  at points  $E$  and  $F$  ( $A$  and  $E$  lie on the same semiplane with respect to  $BC$ ). The line passing through  $E$  and perpendicular to  $FI$  meets  $AB$  and  $AC$  at points  $P$  and  $Q$  respectively. Find the value of  $\angle PIQ$ .

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