
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 that point \( A \) is the left vertex of the hyperbola \( x^{2} - y^{2} = 1 \), and points \( B \) and \( C \) lie on the right branch of the hyperbola such that \( \triangle ABC \) is an equilateral triangle, find the area of \( \triangle ABC\).

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