
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 $H: x^{2}=2py\left(p \gt 0\right)$ with focus $F$, a line passing through the point $\left(0,1\right)$ at a 45-degree angle intersects $H$ at points $A$ and $B$, where $|AB|=8\sqrt{3}$.<br/>$(1)$ Find the equation of the parabola $H$;<br/>$(2)$ Let the equation of the line $l$ be $y=\frac{1}{2}x+n(n≠0)$, and $l$ intersects $H$ at points $C$ and $D$. If a circle $G$ with diameter $CD$ passes through point $F$, find the area of circle $G$.

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