
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]


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Given the function $f(x) = 2\sin\omega x\cos\omega x + \cos2\omega x$ ($\omega > 0$), and the smallest positive period of $f(x)$ is $\pi$,
(1) Find the interval where the function $f(x)$ is monotonically increasing;
(2) If $f\left( \frac {a}{2} - \frac {\pi}{8} \right) = \frac {\sqrt {2}}{3}$ and $f\left( \frac {\beta}{2} - \frac {\pi}{8} \right) = \frac {2 \sqrt {2}}{3}$, with $\alpha, \beta \in \left(-\frac {\pi}{2}, \frac {\pi}{2}\right)$, find the value of $\cos(\alpha + \beta)$.

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