
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 parabola $C$ is the center of the hyperbola $D$: $\frac {y^{2}}{2}-x^{2}= \frac {1}{3}$. The focus of parabola $C$ and the focus of hyperbola $D$ are the same.  
$(1)$ Find the equation of parabola $C$;  
$(2)$ If point $P(t,1)$ $(t > 0)$ is a fixed point on parabola $C$, and $A$, $B$ are two moving points on parabola $C$ with $PA \perp PB$, determine whether line $AB$ passes through a fixed point. If yes, find the fixed point; if no, explain why.

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