
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 its right focus at $F(c,0)$, and a point $P$ on the ellipse $C$. If the maximum and minimum values of $|PF|$ are $2+\sqrt{3}$ and $2-\sqrt{3}$, respectively.

(Ⅰ) Find the equation of the ellipse $C$;

(Ⅱ) Let a line $l$ that does not pass through the origin intersect the ellipse $C$ at points $P$ and $Q$. If the slopes of lines $OP$, $PQ$, and $OQ$ form a geometric sequence, find the maximum area of $\triangle OPQ$.

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