
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 $f(x)=4\sin x\sin \left(x+\frac{\pi }{3}\right)$, in $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively.

(Ⅰ) When $x\in \left[0,\frac{\pi }{2}\right]$, find the range of values for the function $f(x)$;

(Ⅱ) If for any $x\in \mathbb{R}$, $f(x)\leqslant f(A)$ holds, and $b=2$, $c=4$, with point $D$ being the midpoint of side $BC$, find the length of $AD$.

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