
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 eccentricity $\frac{\sqrt{2}}{2}$ and left focus $F_{1}(-1,0)$. The line $l:y=\frac{1}{2}(x+2)$ intersects the ellipse $C$ at two distinct points $A$ and $B$.  
$(1)$ Find the equation of the ellipse $C$.  
$(2)$ Find the area of $\triangle F_{1}AB$.

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