
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]


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Given the equation $\sin (\pi-x)+\sin \left( \frac {\pi}{2}+x\right)=m$ has two real roots $x_{1}$, $x_{2}$ in the interval $[0,2\pi)$, and $|x_{1}-x_{2}|\geqslant \pi$, determine the range of the real number $m$.

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