
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
If $A\geqslant 0$, we say that $A$ has "non-negativity", and when $A=0$, $A$ reaches the minimum value of $0$.  
$(1)$ The algebraic expressions that have non-negativity are ______.  
① $x^{2}-1$;  
② $\left(x-1\right)^{2}$;  
③ $x^{2}+1$;  
④ $|3x+2y|$;  
⑤ $-\left(x-3\right)^{2}$;  
$(2)$ If $A=x^{2}-2x+1$, then when $x=$______, $A$ reaches the minimum value of ______;  
$(3)$ Given $x^{2}+y^{2}=1+xy$, find the maximum value of the algebraic expression $\left(x-3y\right)^{2}+4\left(y+x\right)\left(x-y\right)$.

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