
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 Cartesian coordinate system, the coordinates of points $A$ and $B$ are $A\left(0,a\right)$ and $B\left(0,b\right)$, respectively, and $\sqrt{a+6}+|12-b|=0$.  
$(1)$ Find the square root of $b-a$;  
$(2)$ If there is a point $C$ on the positive half of the $x$-axis, and the area of $\triangle ABC$ is $27$, find the coordinates of point $C$;  
$(3)$ If a line $MN$ parallel to the $y$-axis passes through point $C$ in $(2)$, does there exist a point $D$ on the line $MN$ such that the area of $\triangle ACD$ is $\frac{1}{9}$ of the area of $\triangle ABC$? If yes, find the coordinates of point $D$; if not, explain the reason.

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