
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 the function $f(x)=x^2-2ax+1$, if the range of $a$ for which $f(x)$ has no real roots is the set $A$, and the range of $a$ for which $f(x)$ is not a monotonic function on the interval $(m, m+3)$ is the set $B$,

$(1)$ Find $A$ and $B$;  
$(2)$ If $x \in A$ is a sufficient but not necessary condition for $x \in B$, find the range of $m$.

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