
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 that point $P$ lies on the circle $C: x^{2}-4x+y^{2}=0$, with its abscissa being $1$, and the parabola $x^{2}=-2py\left(p \gt 0\right)$ passes through point $P$, find the equation of the directrix of the parabola.

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