
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
A circle with the origin \\(O\\) as its center intersects the parabola and its directrix \\(y^{2}=4x\\) at points \\(A\\), \\(B\\), and \\(C\\), \\(D\\), respectively. If \\(|AB|=|CD|\\), then the equation of circle \\(O\\) is \_\_\_\_\_\_.

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