
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
Let the function $f(x) = \cos(2x + \frac{\pi}{3}) + \sqrt{3}\sin x\cos x$.

(I) Find the intervals of monotonic decrease for the function $f(x)$ on the interval $\left[-\frac{2\pi}{3}, \frac{\pi}{3}\right]$.

(II) If $\triangle ABC$ satisfies $f(B) = -\frac{1}{18}$, $AC = 2\sqrt{5}$, and $BC = 6$, find the length of $AB$.

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