
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 $\overrightarrow{a}=(1,2)$ and $\overrightarrow{b}=(-3,2)$, for what value of $k$ does
(1) $k \overrightarrow{a}+ \overrightarrow{b}$ and $\overrightarrow{a}-3 \overrightarrow{b}$ are perpendicular?
(2) $k \overrightarrow{a}+ \overrightarrow{b}$ and $\overrightarrow{a}-3 \overrightarrow{b}$ are parallel? When they are parallel, are they in the same or opposite direction?

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