
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 that the function $f(x) = \sqrt{3}\sin\omega x - 2\sin^2\left(\frac{\omega x}{2}\right)$ ($\omega > 0$) has a minimum positive period of $3\pi$,
(I) Find the maximum and minimum values of the function $f(x)$ on the interval $[-\pi, \frac{3\pi}{4}]$;
(II) In $\triangle ABC$, where $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively, and $a < b < c$, with $\sqrt{3}a = 2c\sin A$, find the measure of angle $C$;
(III) Under the conditions of (II), if $f\left(\frac{3}{2}A + \frac{\pi}{2}\right) = \frac{11}{13}$, find the value of $\cos B$.

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