
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 $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ is $\frac{\sqrt{3}}{2}$, and the point $P(2, \sqrt{3})$ is on the ellipse.
(1) Find the equation of the ellipse $C$;
(2) Suppose the left and right vertices of the ellipse are $A$ and $B$, respectively. A moving straight line passing through the point $Q(2,0)$ intersects the ellipse at points $M$ and $N$. The lines $AN$ and $BM$ intersect at point $G$. Find the value of the abscissa of point $G$.

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