
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 a parabola $C$: $y^{2}=2px (p > 0)$ passes through point $A(1,-2)$.
(I) Find the equation of the parabola $C$ and its axis;
(II) Does there exist a line $L$ parallel to $OA$ ($O$ is the origin) such that line $L$ intersects with parabola $C$ and the distance between line $OA$ and $L$ is equal to $\frac{\sqrt{5}}{5}$? If it exists, find the equation of line $L$; if not, explain the reason.

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