
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 a right triangle rectangle  $ABC$  such that  $AB = AC$ ,  $M$  is the midpoint of   $BC$ . Let  $P$  be a point on the perpendicular bisector of   $AC$ , lying in the semi-plane determined by  $BC$  that does not contain  $A$ . Lines  $CP$  and  $AM$  intersect at  $Q$ . Calculate the angles that form the lines  $AP$  and  $BQ$ .

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