
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
Let the set $A=\{(x,y)|\frac{m}{2}\leqslant {(x-2)}^{2}+{y}^{2}\leqslant {m}^{2},x,y\in \mathbb{R}\}$, $B=\{(x,y)|2m\leqslant x+y\leqslant 2m+1,x,y\in \mathbb{R}\}$. If $A\cap B\neq \varnothing$, then the range of the real number $m$ is \_\_\_\_\_\_.

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