
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 $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$) passes through the point $(2,0)$, and the eccentricity of the ellipse $C$ is $\frac{1}{2}$.
(I) Find the equation of ellipse $C$.
(II) If a moving point $P$ lies on the line $x=-1$, construct a line through $P$ that intersects the ellipse $C$ at points $M$ and $N$, such that $P$ is the midpoint of the segment $MN$. Then construct a line $l$ perpendicular to $MN$ through $P$. Determine whether line $l$ always passes through a fixed point; if so, find the coordinates of this point, otherwise explain why not.

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