
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 plane $(xOy)$, the hyperbola $(E)$: $\frac{x^{2}}{a^{2}}-y^{2}=1 (a > 0)$ has left and right foci $F\_1$ and $F\_2$, respectively, with an eccentricity of $\frac{2\sqrt{3}}{3}$. The line $(l)$: $y=x-2$ intersects the right branch of the hyperbola at points $A$ and $B$.
(1) Find the equation of hyperbola $(E)$;
(2) Find the area of triangle $ABF\_1$.

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