
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 $f(x)= \frac {3}{2}\sin \omega x+ \frac { \sqrt {3}}{2}\cos \omega x$ where $\omega > 0$.  
$(1)$ If the smallest positive period of the function $f(x)$ is $3\pi$, find the value of $\omega$, and determine the intervals of monotonic increase for the function $f(x)$;  
$(2)$ If $\omega=2$, $0 < \alpha < \pi$, and $f(\alpha)= \frac {3}{2}$, find the value of $\alpha$.

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