
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 $f(x) = 2\cos^2\omega x + \sin(2\omega x - \frac{\pi}{6})$ ($\omega > 0$).

(1) If the real numbers $x_0$ and $x_0 + \frac{\pi}{2}$ are two adjacent zeros of the function $y = f(x) - 1$, find the value of $\omega$.

(2) In $\triangle BAC$, if $f(\frac{A}{4}) = 2$, $\angle B > \angle C$, $BC = \sqrt{21}$, and the area of $\triangle ABC = \sqrt{3}$, with $O$ being the circumcenter of $\triangle ABC$, find the value of $\overrightarrow{AO} \cdot \overrightarrow{BC}$. (Use the value of $\omega$ found in part (1).)

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