
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 $|\overrightarrow{a}|=4$, $|\overrightarrow{b}|=8$, and the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $120^{\circ}$.  
$(1)$ Calculate: $|\overrightarrow{a}+\overrightarrow{b}|$  
$(2)$ For what value of $k$, does $(\overrightarrow{a}+2\overrightarrow{b})\bot (k\overrightarrow{a}+\overrightarrow{b})$ hold?

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