
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 "whole thinking" is an important thinking method in solving high school mathematics problems, and it is widely used in the simplification and evaluation of polynomials. For example, $4a-2a+a=\left(4-2+1\right)a=3a$. Similarly, if we consider $\left(x+y\right)$ as a whole, then $4\left(x+y\right)-2\left(x+y\right)+\left(x+y\right)=\left(4-2+1\right)\left(x+y\right)=3\left(x+y\right)$. Please follow the above problem-solving method to complete the following questions:  
【Try to Apply】  
$(1)$ Consider $\left(x-y\right)^{2}$ as a whole, combine $3\left(x-y\right)^{2}-6\left(x-y\right)^{2}+2\left(x-y\right)^{2}=$______;  
$(2)$ Given $a^{2}-2b=4$, find the value of $3a^{2}-6b-21$;  
【Further Exploration】  
$(3)$ Given $a-5b=3$, $5b-3c=-5$, $3c-d=10$, find the value of $\left(a-3c\right)+\left(5b-d\right)-\left(5b-3c\right)$.

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