
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 tangent to the parabola $y^2 = 2x$ ($y > 0$), its axis, and the $x$-axis. Determine the equation of the circle.

A) $x^2 + y^2 - x - 2y - \frac{1}{4} = 0$
B) $x^2 + y^2 + x - 2y + 1 = 0$
C) $x^2 + y^2 - x - 2y + 1 = 0$
D) $x^2 + y^2 - x - 2y + \frac{1}{4} = 0$

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