
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 the focus F, and the right focus of the ellipse $\frac {x^{2}}{4}+ \frac {y^{2}}{3}=1$ coincides with F.
(1) Find the equation of the parabola C, and the equation of its directrix;
(2) Let P(1,2). Is there a line l parallel to OP (where O is the origin) such that line l intersects with parabola C, and the distance between line OP and l is $\frac { \sqrt {5}}{5}$? If such a line exists, find the equation of line l; if not, explain why.

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