
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 $\overrightarrow {a} = (1, \sqrt {3})$ and $\overrightarrow {b} = (3, m)$.  
(I) If $\overrightarrow {a} \perp \overrightarrow {b}$, find $|\overrightarrow {b}|$.  
(II) If the angle between vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ is $\frac {\pi}{6}$, find the value of the real number $m$.

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