
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 $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with its right focus $F$ having coordinates $(1, 0)$, and the sum of the distances from any point on the ellipse to the two foci is $4$.
(I) Find the standard equation of the ellipse $C$.
(II) A line $l$ passing through the right focus $F$ intersects the ellipse $C$ at points $P$ and $Q$. Let $Q'$ be the reflection of point $Q$ about the $x$-axis. Determine if $\triangle FPQ'$ has a maximum area. If it exists, find the maximum value; if not, explain the reason.

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