
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 eccentricity $\frac{1}{2}$, and the distance from one endpoint of its minor axis to its left focus is $2$. The line $l$: $y=kx$ intersects the ellipse $C$ at points $M$ and $N$, and $P$ is a point on the ellipse $C$ different from $M$ and $N$.

(Ⅰ) Find the equation of the ellipse $C$;  
(Ⅱ) If the slopes of lines $PM$ and $PN$ both exist, determine whether the product of the slopes of $PM$ and $PN$ is a constant value. If yes, find this constant value; if not, explain why;  
(Ⅲ) Find the maximum area of $\triangle PMN$.

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