
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 sets $A=\{x|x^{2}-3x+2=0\}$, $B=\{x|x^{2}-ax+\left(a-1\right)=0\}$, $C=\{x|x^{2}-mx+2=0\}$.  
$(1)$ Proposition $p$: "$\forall x\in B$, $x\in A$", if proposition $p$ is a true proposition, find the value of $a$;  
$(2)$ If "$x\in A$" is a necessary condition for "$x\in C$", find the range of values for $m$.

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