
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 the point $(2,0)$, and the eccentricity of the ellipse $C$ is $\frac{1}{2}$.

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

(2) If the moving point $P$ is on the line $x=-1$, the line passing through point $P$ intersects the ellipse $C$ at points $M$ and $N$, and $P$ is the midpoint of the line segment $MN$. Then, a line $l$ is drawn through point $P$ and perpendicular to $MN$. Determine whether line $l$ always passes through a fixed point. If it does, find the coordinates of that fixed point. If it doesn't, explain the reason.

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