
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
On the coordinate plane, set \( A = (-1, 0) \), \( B = (1, 0) \), and \( P = (0, t) \) where \( 0 \leq t \leq 1 \). As \( t \) varies, \( C \) is a variable point such that \( P \) is the circumcenter of \( \triangle ABC \). Points which are possible positions of \( C \) are colored red. Find the total area of the red region.

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