
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 $y^{2}=2px\left(p \gt 0\right)$ with the focus $F\left(4,0\right)$, a line $l$ passing through $F$ intersects the parabola at points $M$ and $N$. Find the value of $p=$____, and determine the minimum value of $\frac{{|{NF}|}}{9}-\frac{4}{{|{MF}|}}$.

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