
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 $f(x) = 2\sin\left(\frac{ωx}{2}\right)\left(\sqrt{3}\cos\left(\frac{ωx}{2}\right)-\sin\left(\frac{ωx}{2}\right)\right)$ where $ω > 0$, and its smallest positive period is $3π$.  
(I) Find the value of $ω$ and the maximum and minimum values of $f(x)$ in the interval $[-π, \frac{3π}{4}]$;  
(II) Knowing that $a$, $b$, and $c$ are the side lengths opposite the angles $A$, $B$, $C$ of triangle $ABC$, respectively, $a=2\sqrt{3}$, $c=4$, and $f\left(\frac{3}{2}A\right)=1$, find $b$ and the area of triangle $ABC$.

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