
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 half of the $x$-axis, and the circle is tangent to both line $l$ and the $y$-axis.

(Ⅰ) Find the equation of the circle;

(Ⅱ) The line $l': mx+y+\frac{1}{2}m=0$ intersects the circle $C$ at points $A$ and $B$, and $|AB|= \sqrt{3}$. Find the value of $m$.

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