
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  $ABCD$  be a square with side length  $100$ . Denote by  $M$  the midpoint of  $AB$ . Point  $P$  is selected inside the square so that  $MP = 50$  and  $PC = 100$ . Compute  $AP^2$ .

*Based on a proposal by Amogh Gaitonde*

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