
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
Point $A$ and $B$ are two adjacent intersection points of the line $y=0$ and the graph of the function $f(x)=2\cos^2 \frac{\omega x}{2}+\cos(\omega x+ \frac{\pi}{3})-1$ ($\omega > 0$), and $|AB|= \frac{\pi}{2}$.  
(Ⅰ) Find the value of $\omega$;  
(Ⅱ) In the acute triangle $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively. If $f(A)=-\frac{3}{2}$, $c=3$, and the area of $\triangle ABC$ is $3\sqrt{3}$, find the value of $a$.

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