
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 square  $ABCD$  of diagonals  $AC$  and  $BD$ , we call  $O$  at the center of the square. A square  $PQRS$  is constructed with sides parallel to those of  $ABCD$  with  $P$  in segment  $AO, Q$  in segment  $BO, R$  in segment  $CO, S$  in segment  $DO$ .  If area of  $ABCD$  equals two times the  area of  $PQRS$ ,  and  $M$  is the midpoint of the  $AB$  side, calculate the measure of the angle  $\angle AMP$ . 

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