
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
Reading material: We know that $4x-2x+x=(4-2+1)x=3x$. Similarly, if we consider $(a+b)$ as a whole, then $4(a+b)-2(a+b)+(a+b)=(4-2+1)(a+b)=3(a+b)$. The "whole concept" is an important thinking method in solving problems in secondary school teaching, and it is widely used in the simplification and evaluation of polynomials.  
Try to apply:  
$(1)$ Considering $\left(a-b\right)^{2}$ as a whole, the result of combining $3\left(a-b\right)^{2}-6\left(a-b\right)^{2}+2\left(a-b\right)^{2}$ is ______;  
$(2)$ Given $x^{2}-2y=4$, find the value of $3x^{2}-6y-21$;  
Expand exploration:  
$(3)$ Given $a-2b=6$, $2b-c=-8$, $c-d=9$, find the value of $\left(a-c\right)+\left(2b-d\right)-\left(2b-c\right)$.

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