
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
After learning the complete square formula, we further explore the application of the formula.  
$(1)$ If $ab=30$, $a+b=10$, then the value of $a^{2}+b^{2}$ is ______.  
$(2)$ If $y$ satisfies $\left(40-y\right)\left(y-20\right)=50$, find the value of $\left(40-y\right)^{2}+\left(y-20\right)^{2}$.  
Read the following solution and solve the corresponding problems.  
Solution: Let $40-y=a$, $y-20=b$,  
then $a+b=\left(40-y\right)+\left(y-20\right)=20$,  
$ab=\left(40-y\right)\left(y-20\right)=50$.  
In this way, we can use the method of $(1)$ to find the value.  
Use the above method to solve the following problems:  
① If $x$ satisfies $\left(40-x\right)\left(x-20\right)=-10$, find the value of $\left(40-x\right)^{2}+\left(x-20\right)^{2}$.  
② If $x$ satisfies $\left(30+x\right)\left(20+x\right)=10$, find the value of $\left(30+x\right)^{2}+\left(20+x\right)^{2}$.

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