
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 that the solution set of the inequality $x^{2}-2mx+m+2\leqslant 0$ for $x$ is $M$.  
$(1)$ When $M$ is an empty set, find the range of values for $m$;  
$(2)$ Under the condition in (1), find the minimum value of $f(m)=\frac{{{m^2}+2m+5}}{{m+1}}$;  
$(3)$ When $M$ is not an empty set and $M\subseteq \left[1,4\right]$, find the range of real numbers $m$.

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