
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 that the center of the ellipse $C$ is at the origin and its focus is on the $x$-axis. The eccentricity of the ellipse is $\frac{1}{2}$, and one of its vertices is the focus of the parabola $x^2 = -4 \sqrt{3} y$.

(1) Find the standard equation of the ellipse $C$;

(2) If the line $l$ passing through the point $P (2, 1)$ is tangent to the ellipse $C$ at the point $M$ in the first quadrant, find the equation of the line $l$ and the coordinates of the point $M$.

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