
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 $f(x)=\sin\left(\omega x-\frac{\pi}{6}\right)+\frac{1}{2}$, where $\omega > 0$ and $x \in \mathbb{R}$, and $f(\alpha)=-\frac{1}{2}$, $f(\beta)=\frac{1}{2}$. If the minimum value of $|\alpha-\beta|$ is $\frac{3\pi}{4}$, determine the interval of monotonic increase for the function.

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