
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
The equation of circle $M$ is given by $(x-h)^{2} + (y-k)^{2} = r^{2}$, where $(h,k)$ is the center and $r$ is the radius. The center of circle $M$ is on the $x$-axis and to the right of the line $x = -2$. Find the equation of circle $M$ given that its chord formed by the intersection of the circle and the line $x = -2$ has a length of $2\sqrt{3}$ and is tangent to the line $2x - \sqrt{5}y - 4 = 0$.

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