
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 left and right foci $F_{1}$ and $F_{2}$, respectively. Let point $A(0,b)$ be in $\triangle AF_{1}F_{2}$, where $∠{F}_{1}A{F}_{2}=\frac{π}{2}$ and the perimeter is $2+2\sqrt{2}$.<br/>$(1)$ Find the equation of the ellipse $C$;<br/>$(2)$ A line $l$ is drawn through the left focus $F_{1}$ at an angle of $\frac{π}{3}$ with the $x$-axis, intersecting the ellipse $C$ at points $M$ and $N$. Find the area of $\triangle OMN$.

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