
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 $f(x)=4\sin x\sin (x+\dfrac{\mathrm{ }\!\!\pi\!\!\text{ }}{3})$, in $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively.

(1) Find the range of the function $f(x)$ when $x\in [0,\dfrac{\mathrm{ }\!\!\pi\!\!\text{ }}{2}]$;

(2) If for any $x\in R$, $f(x)\leqslant f(A)$, $b=2$, $c=4$, point $D$ is the midpoint of side $BC$, find the length of $AD$.

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