
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 $C: y^2 = -4x$.
(I) Write the coordinates of the focus of parabola $C$, the equation of the directrix, and the distance from the focus to the directrix;
(II) Line $l$ passes through the fixed point $P(1, 2)$ with a slope of $k$. For what value(s) of $k$ does line $l$ have exactly one common point with the parabola; two common points; no common points.

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