
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 the parabola \( C: y=\frac{1}{2} x^{2} \) and the circle \( D: x^{2}+\left(y-\frac{1}{2}\right)^{2}=r^{2}(r>0) \) have no common points. From a point \( A \) on the parabola \( C \), two tangents are drawn to the circle \( D \), with points of tangency \( E \) and \( F \). As point \( A \) moves along the parabola \( C \), the region formed by all points not lying on the line \( EF \) is denoted as region \( \Omega \). Determine the range of the area of region \( \Omega \).

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