
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
Given a triangle $\triangle ABC$ with vertices at $A(0,1)$, $B(2,1)$, and $C(3,4)$:
1. Find the equation of the circumcircle of $\triangle ABC$.
2. If one endpoint $N$ of a line segment $MN$ is at $(6,2)$ and the other endpoint $M$ moves along the circumcircle of $\triangle ABC$, find the equation of the locus of the midpoint $P$ of the line segment $MN$.

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