
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 $E$: $y^{2}=2px(p > 0)$ whose directrix intersects the $x$-axis at point $k$, and through point $k$, two tangents are drawn to the circle $C$: $(x-5)^{2}+y^{2}=9$ at points $M$ and $N$, with $|MN|=3 \sqrt {3}$.  
$(1)$ Find the equation of the parabola $E$;  
$(2)$ If line $AB$ passes through a fixed point $Q(2,0)$ and intersects the parabola $E$ at points $A$ and $B$, and a perpendicular line to $AB$ through the fixed point $Q$ intersects the parabola at points $G$ and $D$, find the minimum value of the area of quadrilateral $AGBD$.

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