
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 is defined by its center \( C \), which is determined by the position vector \(\overrightarrow{OC} = \vec{c}\), and radius \( R \). Write the equation of the circle in both vector and coordinate forms, given the coordinates of point \( C \) are \( (a, b) \). How do these equations appear depending on the position of the center: a) the center lies on the \( O_x \) axis; b) the center lies on the \( O_y \) axis; c) the center coincides with the origin \( O \), and also in the case when the circle passes through the origin \( O \)?

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