
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 $C$: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$ passing through point $A(1,\frac{\sqrt{2}}{2})$, where point $F(1,0)$ is the right focus of ellipse $C$.  
$(1)$ Find the equation of ellipse $C$;  
$(2)$ Through point $F(1,0)$, draw two lines $l_{1}$ and $l_{2}$ with non-zero slopes that are perpendicular to each other. Line $l_{1}$ intersects the ellipse at points $A_{1}$ and $B_{1}$, while line $l_{2}$ intersects the ellipse at points $A_{2}$ and $B_{2}$. Find the minimum value of the area $S$ of the quadrilateral $A_{1}A_{2}B_{1}B_{2}$.

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