
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
(1) The vertices of triangle $ABC$ are $A(5, 1)$, $B(7, -3)$, and $C(2, -8)$. Find the equation of its circumcircle.
(2) The vertices of triangle $ABC$ are $A(0, 0)$, $B(5, 0)$, and $C(0, 12)$. Find the equation of its incircle.

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