
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
An ellipse has the equation $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ ($a > b > 0$) with one of its vertices at $A(0, 2)$ and eccentricity $e = \frac{\sqrt{6}}{3}$.
(1) Find the equation of the ellipse.
(2) A line $l$ intersects the ellipse at two distinct points $M$ and $N$, with $P(2, 1)$ being the midpoint of $MN$. Find the equation of line $l$.

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