
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 $0\leq x\leq \frac{\pi}{2}$, find the range of the function $f(x)$.
$(2)$ Suppose $\triangle ABC$ has angles $A$, $B$, $C$ opposite to sides $a$, $b$, $c$ respectively. Given that $A$ is an acute angle and $f(A)= \frac{\sqrt{3}}{2}$, $b=2$, $c=3$, find the value of $\cos(A-B)$.

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