
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
In the rectangular coordinate system $(xOy)$, an ellipse $(C)$ is defined by the equation $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ with an eccentricity of $\frac{\sqrt{3}}{2}$. The line $y = x$ intersects the ellipse $(C)$ at points $E$ and $F$, and the line $y = -x$ intersects the ellipse $(C)$ at points $G$ and $H$. The area of the quadrilateral $EHFG$ is $\frac{16}{5}$.

(1) Find the equation of the ellipse $(C)$.

(2) A line $l_1$ passing through the left vertex $A$ of the ellipse $(C)$ intersects the ellipse $(C)$ at another point $P$. Another line $l_2$ passing through the point $A$ and perpendicular to $l_1$ intersects the ellipse $(C)$ at another point $Q$. Determine whether the line $PQ$ passes through a fixed point on the $x$-axis as the slope of the line $l_1$ varies. If it does, find the coordinates of the fixed point; if not, explain why.

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