
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 $\overrightarrow{a} = (3, -1)$, $\overrightarrow{b} = (1, k)$, and $\overrightarrow{a} \perp \overrightarrow{b}$;  
(1) Find the value of $k$;  
(2) Find the angle between $\overrightarrow{a} + \overrightarrow{b}$ and $\overrightarrow{a} - \overrightarrow{b}$.

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