
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 $F$ is the right focus of the ellipse $C: \frac{x^{2}}{4}+\frac{y^{2}}{3}=1$, and the ratio of the distance from any point $P$ on the ellipse $C$ to the point $F$ and the distance from $P$ to the line $l: x=m$ is $\frac{1}{2}$:

(1) Find the equation of the line $l$.

(2) Let $A$ be the left vertex of the ellipse $C$. A line passing through the point $F$ intersects the ellipse $C$ at points $D$ and $E$. The lines $AD$ and $AE$ intersect the line $l$ at points $M$ and $N$, respectively. Does the circle with diameter $MN$ always pass through a fixed point? If so, find the coordinates of the fixed point; if not, provide an explanation.

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