
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
Synthesis and Exploration  
[Reading Comprehension] "Holistic Thinking" is an important mathematical thinking method, widely used in the simplification and evaluation of polynomials.  
For example, $4x-2x+x=\left(4-2+1\right)x=3x$. Similarly, if we consider $\left(a-b\right)$ as a whole, then $4\left(a-b\right)-2\left(a-b\right)+\left(a-b\right)=\left(4-2+1\right)\left(a-b\right)=3\left(a-b\right)$.  
[Try to Apply] Based on the reading content, use "holistic thinking" to answer the following questions:  
$(1)$ Simplify the result of $8\left(a+b\right)+6\left(a+b\right)-2\left(a+b\right)$ is ______.  
$(2)$ Simplify and evaluate, $9\left(x+y\right)^{2}+3\left(x+y\right)+7\left(x+y\right)^{2}-7\left(x+y\right)$, where $x+y=\frac{1}{2}$.  
[Exploration]  
$(3)$ If $x^{2}-2y=4$, find the value of $-3x^{2}+6y+2$.

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