
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 parabola $C$: $y^{2}=2px(p > 0)$ passes through point $A(1,-2)$;  
$(1)$ Find the equation of the parabola $C$ and the equation of its directrix;  
$(2)$ Is there a line $l$ parallel to $OA$ ($O$ is the origin) such that line $l$ has common points with the parabola $C$, and the distance between line $OA$ and $l$ equals $\dfrac{\sqrt{5}}{5}$? If it exists, find the equation of line $l$ and explain the reason.

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