
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 cartesian coordinate system $(xOy)$ on a plane, the curve $G$ is defined as $y=\frac{x^2}{2}+\frac{a}{2}x-a^2 (x\in\mathbb{R})$, where $a$ is a constant.

1. If $a\neq 0$, the graph of curve $G$ intersects both axes at three points. Find the general equation of the circle $C$ that passes through these three points.
2. Under the condition of part 1, find the trajectory equation of the circle center $C$.
3. If $a=0$, given a point $M(0,3)$, there exists a fixed point $N$ (different from point $M$) on the $y$-axis such that for any point $P$ on the circle $C$, the ratio $\frac{|PN|}{|PM|}$ is a constant. Find the coordinates of all points $N$ that satisfy the condition and the constant.

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