
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 $f(x)=2\sqrt{3}\sin\frac{{ωx}}{2}\cos\frac{{ωx}}{2}-2\cos^2\frac{{ωx}}{2}+1$, where $0 < \omega < 4$, and $f(\frac{π}{6})=1$. Find the value of $\omega$ and the interval where the function $f(x)$ is monotonically increasing. Also, find the minimum and maximum values of the function $f(x)$ on the interval $[0, \frac{π}{2}]$.

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