
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 $\overrightarrow {a}=(\sin \frac {x}{2} , \sqrt {3})$ and $\overrightarrow {b}=(\cos \frac {x}{2} , \frac {1}{2}-\cos^{2} \frac {x}{2})$, where $x\in(0, \frac {\pi}{2})$\.
(I) If $\overrightarrow {a}$ is perpendicular to $\overrightarrow {b}$, find the value of $\tan x$;
(II) Consider the function $f(x) = \overrightarrow {a} \cdot \overrightarrow {b}$, and $f(\alpha) = \frac {1}{3}$, find the value of $\sin \alpha$.

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