
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 hyperbola $C$: $\frac{x^2}{4} - y^2 = 1$, where point $A$ is the left vertex of the hyperbola $C$, and point $P$ has coordinates $\left(4,0\right)$.
$(1)$ Through point $P$, draw two parallel lines to the asymptotes of $C$ which intersect the hyperbola $C$ at points $R$ and $S$. Find the equation of the line $RS$.
$(2)$ Through point $P$, draw a line $l$ that intersects the ellipse $\frac{x^2}{4} + y^2 = 1$ at points $D$ and $E$. The lines $AD$ and $AE$ intersect the hyperbola $C$ at points $M$ and $N$, respectively. Determine whether the line $MN$ passes through a fixed point. If it does, find the coordinates of that fixed point; if not, explain why.

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