
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 the functions $f(x)=1+\sin 2x$ and $g(x)=2\cos ^{2}x+m$, if there exists an $x\_0\in\[0, \frac{\pi}{2}\]$ such that $f(x\_0)\geqslant g(x\_0)$, then the range of the real number $m$ is $\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_.$

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