
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 the function $f(x)=\sin \omega x+\cos \omega x$ where $\omega \gt 0$, and $f(\frac{π}{8})=f(\frac{5π}{8})$. If $f(x)$ has exactly one minimum and one maximum in the interval $(\frac{π}{8},\frac{5π}{8})$, then $\omega =\_\_\_\_\_\_$; if $f(x)$ has exactly two zeros in the interval $(\frac{π}{8},\frac{5π}{8})$, then the range of values for $\omega$ is ______.

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