
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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