
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 a line $l: x - \sqrt{3}y + 1 = 0$, and a circle with its center $C$ on the positive x-axis such that the circle is tangent to the line $l$ and the y-axis.

(1) Find the equation of the circle;

(2) If the line $mx + y + \frac{1}{2}m = 0$ intersects the circle $C$ at points $A$ and $B$, and the length of the segment $AB$ is $\sqrt{3}$, find the value of $m$.

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