
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
Points \(A, B, C\) are the vertices of an equilateral triangle. Determine the locus of points \(P\) in the plane of the triangle for which
a) \(PA^{2} + PB^{2} = PC^{2}\)
b) \(PA^{2} + PB^{2} = 2PC^{2}\)

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