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

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