
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 $x-2y+1=0$ intersects the parabola $C:y^{2}=2px\left(p \gt 0\right)$ at points $A$ and $B$, where $|AB|=4\sqrt{15}$.  
$(1)$ Find the value of $p$;  
$(2)$ Let $F$ be the focus of $C$, and $M$ and $N$ be two points on $C$ such that $\overrightarrow{FM}\cdot \overrightarrow{FN}=0$. Find the minimum area of triangle $MFN$.

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