
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 $\overrightarrow{m}=(\sin x, \frac{1}{2})$ and $\overrightarrow{n}=(\cos x,\cos (2x+ \frac{π}{6}))$, let $f(x)= \overrightarrow{m} \cdot \overrightarrow{n}+ \frac{3}{2}$.
(1) Find the interval where the function $f(x)$ is increasing.
(2) In acute triangle $ABC$, let $a$, $b$, and $c$ be the lengths of the sides opposite to angles $A$, $B$, and $C$, respectively. If $f(C)=\frac{3}{2}$ and $c=\sqrt{3}$, find the range of $a-\frac{1}{2}b$.

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