
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
Given $y=x^{2}-2ax+a$.<br/>$(1)$ If the equation $y=0$ has two real roots $x_{1}$ and $x_{2}$, and ${x}_{1}^{2}+{x}_{2}^{2}=6x_{1}x_{2}-3$, find the value of the real number $a$;<br/>$(2)$ If sets $A=\{x|x^{2}+4x=0\}$, $B=\{x|y+a^{2}-a=-\left(4a+2\right)x+1\}$, and $A\cup B=A$, find the range of values for $a$.

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