
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 $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$ with an eccentricity $e= \frac{2 \sqrt{5}}{5}$, the distance between the left vertex $A$ and the right focus $F$ is $AF=2+ \sqrt{5}$.
(1) Find the equation of the ellipse $C$;
(2) A line $l$ with slope $k$ passes through the right focus $F$ and intersects the ellipse $C$ at points $M$ and $N$. $P(2,1)$ is a fixed point. When the area of $\triangle MNP$ is maximum, find the equation of $l$.

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