
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 $\overrightarrow{a}=({3,2,-1})$ and $\overrightarrow{b}=({2,1,2})$.  
$(1)$ Find the cosine value of the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$.  
$(2)$ When $({k\overrightarrow{a}+\overrightarrow{b}})⊥({\overrightarrow{a}-k\overrightarrow{b}})$, find the value of the real number $k$.

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