
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 that $x_{0}= \frac{\pi}{3}$ is an axis of symmetry for the function $f(x)=m\sin \omega x-\cos \omega x$ ($m > 0$), and the smallest positive period of $f(x)$ is $\pi$.  
$(1)$ Find the value of $m$ and the intervals of monotonic increase for $f(x)$;  

$(2)$ Let angles $A$, $B$, $C$ be the three interior angles of $\triangle ABC$, with corresponding sides $a$, $b$, $c$. If $f(B)=2$ and $b= \sqrt{3}$, find the range of values for $a- \frac{c}{2}$.

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