
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 $f(x)=\cos x(\sqrt{3}\sin x-\cos x)+m$ ($m\in \mathbb{R}$), the graph of $y=f(x)$ is translated to the left by $\dfrac{\pi}{6}$ units to obtain the graph of $g(x)$. The minimum value of $y=g(x)$ in the interval $\left[\dfrac{\pi}{4},\dfrac{\pi}{3}\right]$ is $\dfrac{\sqrt{3}}{2}$.

$(1)$ Find the value of $m$;  
$(2)$ In an acute $\triangle ABC$, if $g\left(\dfrac{c}{2}\right)=-\dfrac{1}{2}+\sqrt{3}$, find the range of values for $\sin A+\cos B$.

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