
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 $C: y^{2}=2px\left(p \gt 0\right)$, where the point on the parabola $C$ with an x-coordinate of $1$ is at a distance of $3$ from the focus $F$. <br/>$(Ⅰ)$ Find the equation of the parabola $C$ and its directrix equation; <br/>$(Ⅱ)$ A line $l$ passing through $(-1,0)$ intersects the parabola $C$ at two distinct points $A$ and $B$, intersects the line $x=-4$ at point $E$, and the line $BF$ intersects the line $x=-1$ at point $D$. Does there exist a line $l$ such that $DE$ is parallel to $AF$? If not, explain why; if yes, find the equation of the line $l$.

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