
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 comprehension】According to the law of combining like terms, we have $5x-3x+4x=(5-3+4)x=6x$; similarly, if we consider $(m+n)$ as a whole, then $5(m+n)-3(m+n)+4(m+n)=(5-3+4)(m+n)=6(m+n)$. This problem-solving method is called "whole thinking", and the application of whole thinking is extremely extensive in the simplification and evaluation of polynomials.<br/>【Try to apply】<br/>$(1)$ Considering $\left(a-b\right)^{2}$ as a whole, the result of combining $10\left(a-b\right)^{2}-12\left(a-b\right)^{2}+9\left(a-b\right)^{2}$ is______;<br/>$(2)$ Given $x^{2}-2y=-5$, find the value of $4x^{2}-8y+24$;<br/>【Further exploration】<br/>$(3)$ Given $a-2b=1009\frac{1}{2}$, $2b-c=-2024\frac{2}{3}$, $c-d=1013\frac{1}{6}$, 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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