
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
Find the equations of the following circles:

(1) Given the coordinates of the vertices of $\triangle ABC$ as $A(4,0)$, $B(0,3)$, and $C(0,0)$, find the equation of the circumcircle of $\triangle ABC$.

(2) Find the equation of the circle whose center lies on the $y$-axis, has a radius of $5$, and is tangent to the line $y=6$.

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