
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 ellipse $c: \frac{x^2}{a^2} + \frac{y^2}{3} = 1$ ($a > \sqrt{10}$), its right focus $F$ is on the circle $D: (x-2)^2 + y^2 = 1$. The line $l: x=my+3$ ($m \neq 0$) intersects the ellipse at points $M$ and $N$.  
(Ⅰ) Find the equation of ellipse $C$.  
(Ⅱ) If $\overrightarrow{OM} \perp \overrightarrow{ON}$ ($O$ is the origin), find the value of $m$.  
(Ⅲ) Let the point symmetric to $N$ with respect to the $x$-axis be $N_1$ ($N_1$ and $M$ do not coincide), and the line $N_1M$ intersects the $x$-axis at point $P$. Determine whether the area of $\triangle PMN$ has a maximum value. If it exists, find this maximum value; if not, explain why.

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