
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
Let vector $\overrightarrow {a} = (\cos\theta, \sin\theta)$, and vector $\overrightarrow {b} = \left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$;  
(1) If $\overrightarrow {a} \parallel \overrightarrow {b}$, and $\theta \in (0, \pi)$, find $\theta$;  
(2) If $|3\overrightarrow {a} + \overrightarrow {b}| = |\overrightarrow {a} - 3\overrightarrow {b}|$, find the value of $|\overrightarrow {a} + \overrightarrow {b}|$.

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