
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 line $l\_1$: $x+y-3=0$ intersects the $x$-axis at point $A$, line $l\_2$ has equation $y=2x$, and intersects line $l\_1$ at point $B$. Point $C$ is on the negative half of the $y$-axis, and $AC=2\sqrt{3}$.

(1) Find the area of $\triangle ABC$.

(2) Find the equation of the circumcircle of $\triangle ABC$.

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