
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 $f(x)=\cos(\omega x+\frac{\pi}{3})$ where $\omega\gt 0$ be a function. If the smallest positive period of $f(x)$ is $T$, and the graph of $y=f(x)$ is symmetric about $x=\frac{\pi}{6}$, then when $\omega$ takes its minimum value, $f(\frac{T}{2})$ equals $\_\_\_\_\_\_$.

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