
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 $\overrightarrow{a}=(1,5,-1)$ and $\overrightarrow{b}=(-2,3,5)$.

$(1)$ If $(k \overrightarrow{a}+ \overrightarrow{b})\parallel( \overrightarrow{a}-3 \overrightarrow{b})$, find $k$;  
$(2)$ If $(k \overrightarrow{a}+ \overrightarrow{b})\perp( \overrightarrow{a}-3 \overrightarrow{b})$, find $k$.

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