
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 function $f(x)=\sin(\omega x+\frac{π}{6})(\omega>0)$ is monotonically increasing on the interval $[-\frac{{5π}}{6},\frac{{2π}}{3}]$, and there exists a unique $x_0 \in [0,\frac{{5π}}{6}]$ such that $f(x_{0})=1$, determine the range of $\omega$.

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