
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
In the Cartesian coordinate system $xOy$, the hyperbola $E$: $\frac{x^2}{a^2} - y^2 = 1$ ($a > 0$) has its left and right foci as $F_1$ and $F_2$, respectively, with an eccentricity of $\frac{2\sqrt{3}}{3}$, and a line $l$ passing through the right focus $F_2$ intersects the right branch of the hyperbola at points $A$ and $B$.  
$(1)$ Find the equation of the hyperbola $E$;  
$(2)$ Find the range of the area of $\triangle ABF_{1}$.

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