
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
Let $\overrightarrow{a}$ and $\overrightarrow{b}$ be two non-collinear non-zero vectors.<br/>$(1)$ If $8\overrightarrow{a}+k\overrightarrow{b}$ and $k\overrightarrow{a}+2\overrightarrow{b}$ are collinear, find the value of the real number $k$;<br/>$(2)$ If $|\overrightarrow{a}|=4$, $|\overrightarrow{b}|=3$, and $(2\overrightarrow{a}-3\overrightarrow{b})\cdot(2\overrightarrow{a}+\overrightarrow{b})=61$, find the value of $|\overrightarrow{a}+\overrightarrow{b}|$.

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