
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 $T: y^{2}=2px\left(p \gt 0\right)$, where point $F$ is its focus, the line $l: x=4$ intersects the parabola at points $M$ and $N$, and $O$ is the origin. It is known that $S_{\triangle OMN}=8\sqrt{6}$.  
$(1)$ Find the equation of the parabola $T$.  
$(2)$ For a moving point $E(a,0)$ on the $x$-axis $(a \gt 0)$, two lines perpendicular to each other are drawn, intersecting the parabola $T$ at points $A$, $B$, $C$, and $D$. Points $H$ and $K$ are the midpoints of $AB$ and $CD$ respectively. Find the minimum value of $|HK|$.

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