
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 the line $x+2y-2=0$ passes through the focus of the parabola $C: x^{2}=2py\left(p \gt 0\right)$.
$(1)$ Find the equation of the parabola $C$;
$(2)$ Point $A$ lies on the directrix of the parabola $C$. Two tangent lines of the parabola $C$ passing through point $A$ intersect the $x$-axis at points $M$ and $N$. When the area of triangle $AMN$ is $\frac{{\sqrt{5}}}{2}$, find the coordinates of point $A$.

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