
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
There is a question: "If the value of the algebraic expression $5a+3b$ is $-4$, then what is the value of the algebraic expression $2\left(a+b\right)+4\left(2a+b\right)$?" Tang, who loves to use his brain, solved the problem as follows:  
Original expression $=2a+2b+8a+4b=10a+6b=2\left(5a+3b\right)=2\times \left(-4\right)=-8$  
Tang treated $5a+3b$ as a whole to solve it. Treating expressions as a whole is an important method in solving problems in high school mathematics. Please follow the method above to complete the following questions:  
$(1)$ Given $a^{2}+a=3$, then $2a^{2}+2a+2023=$______;  
$(2)$ Given $a-2b=-3$, find the value of $3\left(a-b\right)-7a+11b+2$;  
$(3)$ Given $a^{2}+2ab=-5$, $ab-2b^{2}=-3$, find the value of the algebraic expression $a^{2}+ab+2b^{2}$.

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