
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
Let the parabola $C: x^2 = 2py$ ($p > 0$) have focus $F$ and directrix $l$. Let $A$ belong to $C$. It is known that the circle with center $F$ and radius $FA$ intersects $l$ at points $B$ and $D$;  
(1) If $\angle BFD = 90^\circ$ and the area of $\triangle ABD$ is $4\sqrt{2}$, find the value of $p$ and the equation of circle $F$;  
(2) If points $A$, $B$, $F$ lie on the same line $m$, and line $n$ is parallel to $m$ and intersects $C$ at only one point, find the ratio of the distances from the origin to lines $m$ and $n$.

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