
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 $\omega\ \ \gt 0$, the function $f(x)=\sin(\omega x+\frac{π}{3})\cdot\sin(\frac{π}{3}-\omega x)$ has a minimum positive period of $\pi$.  
$(1)$ Find the intervals where the function $f(x)$ is monotonically increasing.  
$(2)$ If $\frac{π}{8}＜α＜\frac{5π}{8}$ and $f(α-\frac{π}{8})=\frac{11}{20}$, find the value of $\cos 2\alpha$.

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