
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
$(1)$ Let $M=2x^{2}+1$ and $N=x^{2}+2x-1$. Compare the sizes of $M$ and $N$;<br/>$(2)$ Given conditions $\alpha :2m\leqslant x\leqslant m+1$, condition $\beta :-1\leqslant x\leqslant 1$, if $\alpha $ is a sufficient condition for $\beta $, find the range of real number $m$.

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