
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 eccentricity of the ellipse $C_{1}$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$ is $e= \frac{\sqrt{3}}{2}$, and the right focus of $C_{1}$ coincides with the focus of the parabola $C_{2}$: $y^{2}=4\sqrt{3}x$.
(1) Find the equation of the ellipse $C_{1}$;
(2) Find the value of $k_{1}⋅k_{2}$ where $k_{1}≠k_{2}$ when the line $MN$ is perpendicular to the $y$-axis. The line $MN$ is formed by the intersection of the ellipse $C_{1}$ and two lines passing through the point $P(-2,0)$ with slopes $k_{1}$ and $k_{2}$.

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