
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 circle \(\Omega\) and the parabola \(\Gamma: y^2 = 4x\) have exactly one common point, and the circle \(\Omega\) is tangent to the \(x\)-axis at the focus \(F\) of the parabola \(\Gamma\). Find the radius of the circle \(\Omega\).

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