
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 Cartesian coordinate system $xOy$, the vertex of the parabola $C$ is the center of the hyperbola $D$: $$\frac{y^2}{2} - x^2 = \frac{1}{3}.$$ The focus of the parabola $C$ coincides with the foci of the hyperbola $D$.

(1) Find the equation of the parabola $C$;
(2) If the point $P(t, 1)$ ($t > 0$) is a fixed point on the parabola $C$, and $A$, $B$ are two moving points on the parabola $C$ such that $PA \perp PB$, determine whether the line $AB$ passes through a fixed point. If it does, find the fixed point; if not, explain why.

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