
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 a parabola $y^2 = 2x$ and a point $A(x_0, y_0)$ on the parabola where $x_0 > 2$, two tangents are drawn from point A to the circle $(x-1)^2 + y^2 = 1$, intersecting the y-axis at points B and C. Determine the minimum area of triangle ABC.

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