
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 $C: y^{2}=2px\left(p \lt 0\right)$ passing through the point $A\left(-2,-4\right)$.  
$(1)$ Find the equation of the parabola $C$ and its directrix equation.  
$(2)$ A line passing through the focus of the parabola, making an angle of $60^{\circ}$ with the $x$-axis, intersects the parabola at points $A$ and $B$. Find the length of segment $AB$.

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