
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 of side length  $1$ , and let  $P_1, P_2$  and  $P_3$  be points on the perimeter such that  $\angle P_1P_2P_3 = 90^\circ$  and  $P_1, P_2, P_3$  lie on different sides of the square. As these points vary, the locus of the circumcenter of  $\triangle P_1P_2P_3$  is a region  $\mathcal{R}$ ; what is the area of  $\mathcal{R}$ ?

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