
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  $AB$  be a segment of length  $2$ .  The locus of points  $P$  such that the  $P$ -median of triangle  $ABP$  and its reflection over the  $P$ -angle bisector of triangle  $ABP$  are perpendicular determines some region  $R$ .  Find the area of  $R$ .

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