
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 $C$: $y^{2}=2px(p > 0)$ passes through point $A(1,-2)$.

$(1)$ Find the equation of the parabola $C$ and the equation of its directrix;

$(2)$ If a line $l$ parallel to $OA$ ($O$ is the origin) intersects the parabola $C$ at points $M$ and $N$, and $|MN|=3 \sqrt{5}$, find the area of $\triangle AMN$.

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