
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
If the function $y=\cos \left( \omega x+ \frac{\pi }{6} \right)(\omega \in \mathbb{N}^*) $ has one of its symmetric centers at point $\left( \frac{\pi }{6},0 \right)$, find the minimum value of $\omega$.

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