
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
It is known that the parabola $C: y^2 = 2px$ (where $p > 0$) passes through point $A(2, -4)$.
1. Find the equation of the parabola $C$ and the equation of its directrix $l$.
2. If point $B$ is at $(0, 2)$, find the equation of the line $l$ that passes through point $B$ and has exactly one common point with parabola $C$.

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