
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 vectors $\overrightarrow{a}=(2\sin \omega x,2\cos \omega x)$ and $\overrightarrow{b}=(\sqrt{3}\cos\omega x,-\cos\omega x)$, where the function $f(x)=\overrightarrow{a}\cdot\overrightarrow{b}$ has a minimum positive period of $6\pi$. Find the value of the real number $\omega$. Additionally, given $α,β∈[\frac{π}{6},\frac{2π}{3}]$, where $f(3α)=-\frac{3}{13}$ and $f(3β+3π)=-\frac{11}{5}$, find $\sin(\alpha -\beta)$.

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