
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 $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, its left and right foci are $F_1$ and $F_2$ respectively. Point $P(1, \frac{\sqrt{2}}{2})$ is on the ellipse, and $|PF_1| + |PF_2| = 2\sqrt{2}$.

$(1)$ Find the standard equation of ellipse $C$;  
$(2)$ A line $l$ passing through $F_2$ intersects the ellipse at points $A$ and $B$. Find the maximum area of $\triangle AOB$.

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