
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 $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$ with an eccentricity of $\frac{1}{2}$, passing through point $P(1, \frac{3}{2})$, and $F$ as its right focus.

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

(II) Let line $l$ pass through point $A(4,0)$ intersecting the ellipse at points $M$ and $N$ (with $M$ between $A$ and $N$). If the areas of triangles $AMF$ and $MFN$ are equal, find the equation of line $l$.

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