
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 xOy Cartesian coordinate system, consider the curve G given by the equation $y= \frac{x^2}{2} + \frac{a}{2}x - a^2$ where $x \in \mathbb{R}$ and $a$ is a constant.
(1) If $a\neq 0$, the graph of curve G intersects with the two coordinate axes at three points. Find the general equation of the circle C passing through these three intersection points.
(2) Under the condition in (1), find the equation of the locus of the center of circle C.
(3) If $a=0$ and point M(0,3) is given, there exists a fixed point N (different from point M) on the y-axis such that for any point P on circle C, the ratio $\frac{|PN|}{|PM|}$ is a constant. Determine all possible coordinates for point N and the value of this constant.

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