
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 $\overrightarrow{a}=(\sin (x+ \frac {\pi}{6}),1)$ and $\overrightarrow{b}=(4,4\cos x- \sqrt {3})$  
$(I)$ If $\overrightarrow{a}\perp \overrightarrow{b}$, find the value of $\sin (x+ \frac {4\pi}{3})$;  
$(II)$ Let $f(x)= \overrightarrow{a}\cdot \overrightarrow{b}$, if $\alpha\in[0, \frac {\pi}{2}]$ and $f(\alpha- \frac {\pi}{6})=2 \sqrt {3}$, find the value of $\cos \alpha$.

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