
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 quadratic function $y=x^{2}-(m+2)x+3$,<br/>$(1)$ Let $y=x^{2}-(m+2)x+3$ have a maximum value of $M$ and a minimum value of $N$ in the range $1\leqslant x\leqslant 2$, and $M-N\leqslant 2$. Find the range of real numbers for $m$;<br/>$(2)$ Given the equation $x^{2}-(m+2)x+3=-(2m+1)x+2$ has a solution in the range $0\leqslant x\leqslant 2$, find the range of real numbers for $m$.

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