
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 a focal distance of $2$ and an eccentricity of $\frac{\sqrt{2}}{2}$,  
$(1)$ find the equation of the ellipse $C$;  
$(2)$ let the right focus of the ellipse be $F$, and $K$ be a point on the $x$-axis such that $\overrightarrow{OK} = 2\overrightarrow{OF}$. A line $l$ with a non-zero slope passes through point $K$ and intersects the ellipse at points $P$ and $Q$. Find the maximum value of the area $s$ of $\triangle FPQ$.

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