
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
Let $f(x)=\cos(\omega x+\varphi)$, where $\omega\gt 0$ and $0\lt \varphi\lt \pi$. The smallest positive period of the function is $T$. If $f(T)=\frac{\sqrt{3}}{2}$ and $x=\frac{π}{9}$ is a zero of $f(x)$, then the minimum value of $\omega$ is ____.

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