
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 $\vec{a} =(1, \sqrt{3} )$ and $\vec{b} =(3, m)$.  
(1) If $\vec{a} \perp \vec{b}$, find $|\vec{b}|$;  
(2) If the angle between vectors $\vec{a}$ and $\vec{b}$ 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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