
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. Given the vertices of $\triangle ABC$, $A(4,4)$, $B(5,3)$, and $C(1,1)$, find the equation of the circumcircle of $\triangle ABC$.
2. Find the equation of the circle whose center lies on the $x$-axis and is tangent to the lines $l_{1}: 4x-3y+5=0$ and $l_{2}: 3x-4y-5=0$.

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