
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 a function $f\left(x\right)=x^{2}-2mx+m+2\left(x\in R\right)$, if $f\left(x\right)$ has two zeros and $f\left(x\right)$ is monotonically increasing on $\left[1,+\infty \right)$, then the range of real number $m$ is ______.

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