
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 $A=\{1,2\}$, $B=\{x|x^{2}-2ax+a=0\}$, and $C=\{x|x^{2}-mx+3 \gt 0\}$.  
$(1)$ If the proposition $p$: "$\forall x\in B$, $x\in A$" is true, find the range of real number $a$.  
$(2)$ If "$x\in C$" is a necessary condition for "$x\in A$", find the range of real number $m$.

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