
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 vectors $\overrightarrow {a} = (2\cos x, 2\sin x)$, $\overrightarrow {b} = (\sqrt {3}\cos x, \cos x)$, and the function $f(x) = \overrightarrow {a} \cdot \overrightarrow {b} - \sqrt {3}$, find:  
(1) The smallest positive period and the intervals of monotonic increase for $f(x)$;  
(2) If $f\left( \frac {\alpha}{2} - \frac {\pi}{6}\right) - f\left( \frac {\alpha}{2} + \frac {\pi}{12}\right) = \sqrt {6}$, and $\alpha \in \left( \frac {\pi}{2}, \pi \right)$. Find $\alpha$.

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