
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 $(C)$: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ passing through point $Q(1, \frac{3}{2})$, with an eccentricity of $e = \frac{1}{2}$.

(I) Find the equation of ellipse $(C)$;

(II) Let $A, B$ be the endpoints of the major axis of ellipse $(C)$, and $P$ be a moving point on the ellipse other than $A, B$. The line $l$: $x=4$ intersects with lines $PA, PB$ at points $M, N$ respectively. Moreover, there is a point $E(7, 0)$. Determine whether a circle passing through points $E, M, N$ also passes through another fixed point on the $x$-axis. If it does, find the coordinates of that fixed point. If not, explain why.

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