
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 $\overrightarrow{a} = (1, 0)$, $\overrightarrow{b} = (2, 1)$. Find:  
(1) $|\overrightarrow{a} + 3\overrightarrow{b}|$;  
(2) For what real number $k$ is $k\overrightarrow{a} - \overrightarrow{b}$ parallel to $\overrightarrow{a} + 3\overrightarrow{b}$, and when they are parallel, are they in the same or opposite direction?  
(3) When the vector $k\overrightarrow{a} - \overrightarrow{b}$ is perpendicular to $3\overrightarrow{a} - \overrightarrow{b}$, find the cosine of the angle between the vector $k\overrightarrow{a} - \overrightarrow{b}$ and $\overrightarrow{b}$.

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