
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 one of its foci being $F(3,0)$ and its left vertex $A$ lying on the circle $O$: $x^{2}+y^{2}=12$.
1. Find the equation of the ellipse $C$.
2. The line $l$: $x=my+3 (m\neq 0)$ intersects the ellipse $C$ at points $M$ and $N$. Let $N_1$ be the reflection of point $N$ over the $x$-axis (point $N_1$ is not coincident with point $M$), and let $P$ be the intersection of line $N_1M$ with the $x$-axis. Determine whether the area of $\triangle PMN$ has a maximum value. If it exists, find this maximum value; if not, explain the reason.

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