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

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

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

Candidates should choose to answer either question $(22)$ or $(23)$. If more than one question is answered, only the first one will be graded. Please indicate the question number when answering.

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