
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 that the right focus of the ellipse $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$ is $F(c,0)$, and point $P$ is a moving point 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 the line $l$ that does not pass through the origin intersect the ellipse $C$ at points $P$ and $Q$. If the slopes of the 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}
user