
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 $\overrightarrow{a}=(\sin \omega x,\cos \omega x)$, $\overrightarrow{b}=(\cos \omega x,\sqrt{3}\cos \omega x)$, where $\omega > 0$, and the function $f(x)=\overrightarrow{a}\cdot (\overrightarrow{b}-\frac{\sqrt{3}}{2}\overrightarrow{a})$ has a minimum positive period of $\pi$. Find:<br/>
$(1)$ The intervals where the function $f(x)$ is monotonically increasing.<br/>
$(2)$ In acute triangle $\triangle ABC$, where the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and it satisfies $f(\frac{A}{2})=\frac{\sqrt{3}}{2}$, find the range of $\frac{a}{b}$.

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