
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
In the plane rectangular coordinate system $xOy$, it is known that the moving circle $M$ is tangent to the circle $x^{2}+y^{2}-2x=0$ and tangent to the line $x=-2$. Let the locus of the center of the moving circle $M$ be curve $E$.  
$(1)$ Find the equation of curve $E$;  
$(2)$ Two lines perpendicular to each other pass through point $F(1,0)$ and intersect curve $E$ at points $A$ and $B$, and points $C$ and $D$. Find the minimum value of the area $S$ of quadrilateral $ACBD$.

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