
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 $D: \frac{{y}^{2}}{3} - {x}^{2} = 1$, and the focus of the parabola $C$ is the same as the focus of the hyperbola $D$.  
$(1)$ Find the equation of the parabola $C$;  
$(2)$ If point $P(t,2) (t > 0)$ is a fixed point on the parabola $C$, $A$ and $B$ are two moving points on the parabola $C$, and $PA \perp PB$, determine whether the line $AB$ passes through a fixed point. If yes, find the fixed point; if not, explain the reason.

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