
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
In $\triangle ABC$, the sides $a$, $b$, $c$ are opposite to angles $A$, $B$, $C$ respectively; and $b=4$, $A= \frac {\pi}{3}$, the area $S=2 \sqrt {3}$.  
(Ⅰ) Find the value of $a$;  
(Ⅱ) Let $f(x)=2(\cos C\sin x-\cos A\cos x)$, transform the graph of $f(x)$ by making the x-coordinates of all points half of their original values (the y-coordinates remain unchanged) to get the graph of $g(x)$, find the intervals where $g(x)$ is increasing.

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