
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: $\overrightarrow{a}=(2\cos x,\sin x)$, $\overrightarrow{b}=( \sqrt {3}\cos x,2\cos x)$. Let the function $f(x)= \overrightarrow{a}\cdot \overrightarrow{b}- \sqrt {3}$ $(x\in\mathbb{R})$. Find:  
(1) The smallest positive period of $f(x)$;  
(2) The interval of monotonic increase for $f(x)$;  
(3) If $f( \frac {α}{2}- \frac {π}{6})-f( \frac {α}{2}+ \frac {π}{12})= \sqrt {6}$, and $α\in( \frac {π}{2},π)$, find $α$．

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