
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 the ellipse
$$
C_{1}: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)
$$

with an eccentricity of $\frac{\sqrt{3}}{2}$, the right focus is the center of the circle
$$
C_{2}:(x-\sqrt{3})^{2}+y^{2}=7.
$$

1. Find the equation of the ellipse $C_{1}$.
2. If the line $l$ intersects each of the curves $C_{1}$ and $C_{2}$ at exactly one point, and the point of intersection of the line $l$ and circle $C_{2}$ is denoted as $A$, determine the coordinates of point $A$.

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