
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 $A=\{x|x^{2}-3x+2=0\}$, $B=\{x|x^{2}-ax+a-1=0\}$, $C=\{x|x^{2}+2\left(m+1\right)x+m^{2}-5=0\}$.  
$(1)$ If $A\cup B=A$, find the value of real number $a$;  
$(2)$ If $A\cap C=C$, find the range of real number $m$.

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