
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 $f(x) = \sin \omega x \cdot \cos \omega x - \sqrt{3}\cos^2 \omega x + \frac{\sqrt{3}}{2}$ ($\omega > 0$) have its graph such that the distance between adjacent highest and lowest points is $\sqrt{\pi^2 + 4}$.
(1) Find the value of $\omega$;
(2) If the function $y = f(x + \varphi)$ ($0 < \varphi < \frac{\pi}{2}$) is an odd function, find the intervals of decrease for the function $g(x) = \cos(2x - \varphi)$ in the interval $[0, 2\pi]$.

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