
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 $\overrightarrow{a}=\left(2,-1,-4\right)$ and $\overrightarrow{b}=\left(-1,k,2\right)$.<br/>$(1)$ If ($\overrightarrow{a}-\overrightarrow{b}$) is parallel to ($\overrightarrow{a}+\overrightarrow{b}$), find the value of the real number $k$;<br/>$(2)$ If ($\overrightarrow{a}+3\overrightarrow{b}$) is perpendicular to ($\overrightarrow{a}+\overrightarrow{b}$), find the value of the real number $k$.

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