
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 $\overrightarrow{a}=({1,4,-2})$, $\overrightarrow{b}=({-2,2,4})$.  
$(1)$ If $\overrightarrow{c}=\frac{1}{2}\overrightarrow{b}$, find the value of $cos＜\overrightarrow{a},\overrightarrow{c}＞$;  
$(2)$ If $({k\overrightarrow{a}+\overrightarrow{b}}){∥}({\overrightarrow{a}-3\overrightarrow{b}})$, find the value of the real number $k$.

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