
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 $C$: $y^{2}=2px (p > 0)$, with a point $M(1,m)$ on it, the distance from $M$ to its focus $F$ is $2$.

(I) Find the equation of $C$ and its latus rectum equation.
(II) Given $A$ $(1,-2)$, 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 $OA$ and $L$ equals $\frac{\sqrt{5}}{5}$? If it exists, find the equation of line $L$; if not, explain why.

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