
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
Let point \( C \) be a moving point on the parabola \( y^2 = 2x \). From \( C \), two tangent lines \( CA \) and \( CB \) are drawn to the circle \((x-1)^2 + y^2 = 1\), intersecting the negative half of the \( y \)-axis at \( A \) and the positive half of the \( y \)-axis at \( B \). Find the minimum area of triangle \( \triangle ABC \).

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