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

(1) Find the range of the function $f(x)$ when $0\leqslant x\leqslant \frac{\pi}{2}$.

(2) In triangle $ABC$, let $A$, $B$, $C$ be the angles opposite to sides $a$, $b$, $c$ respectively. Given that $A$ is an acute angle, $b=2$, $a=\sqrt{7}$, and $f(A)=\frac{\sqrt{3}}{2}$, find the value of $\cos(A-B)$.

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