
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 $E$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$ is $\frac{1}{2}$, point $F$ is its right focus, point $A$ is its left vertex, and $|AF|=3$.
(I) Find the equation of ellipse $E$;
(II) A line passing through point $F$ that does not coincide with the $x$-axis intersects ellipse $E$ at two points $B$ and $C$. Lines $AB$ and $AC$ intersect line $l$: $x=4$ at points $M$ and $N$, respectively. Is there a fixed point $Q$ on the $x$-axis such that $\overrightarrow{QM}\cdot \overrightarrow{QN}=0$? If so, find the coordinates of fixed point $Q$; if not, explain the reason.

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