
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 standard equation of parabola $C$ is $y^2 = 2px$ ($p > 0$), $M$ is a moving point on parabola $C$, and $A(a, 0)$ ($a \neq 0$) is a point on its axis of symmetry. The line $MA$ intersects parabola $C$ at another point $N$. When $A$ is the focus of parabola $C$ and line $MA$ is perpendicular to its axis of symmetry, the area of $\triangle MON$ is 18.
(1) Find the standard equation of parabola $C$;
(2) Let $t = \frac{1}{|AM|} + \frac{1}{|AN|}$. If the value of $t$ is independent of the position of point $M$, then point $A$ is called a "stable point". Find all "stable points", or explain why there are none.

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