
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 \( xOy \), given a moving line \( l \) that is tangent to the parabola \( \Gamma: y^{2}=4x \) and intersects the hyperbola \( \Omega: x^{2}-y^{2}=1 \) at one point on each of its left and right branches at points \( A \) and \( B \), find the minimum area of \( \triangle OAB \).

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