
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 $f(x)=2\sin \left(x+ \frac{\pi}{3}\right)\cdot\cos x$.

$(1)$ If $0\leqslant x\leqslant \frac{\pi}{2}$, find the range of the function $f(x)$.

$(2)$ Let $\triangle ABC$ have internal angles $A$, $B$, $C$ corresponding to sides $a$, $b$, $c$ respectively. If $A$ is an acute angle and $f(A)= \frac{\sqrt{3}}{2}$, with $b=2$ and $c=3$, find the value of $\cos (A-B)$.

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