
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
In the Cartesian coordinate plane $(xOy)$, a circle $C$ with center on the $x$-axis and radius $2$ is located to the right of the $y$-axis and tangent to the line $x - \sqrt{3}y + 2 = 0$.
(1) Find the equation of circle $C$;
(2) Determine if there exists a point $M(m, n)$ on circle $C$ such that the line $l: mx + ny = 1$ intersects circle $O: x^2 + y^2 = 1$ at two distinct points $A$ and $B$, and $\triangle OAB$ has the maximum area. If such a point exists, find the coordinates of point $M$ and the corresponding area of $\triangle OAB$. If not, provide an explanation.

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