
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 $|\overrightarrow{a}|=2$, $|\overrightarrow{b}|=1$, $(2\overrightarrow{a}-3\overrightarrow{b})\cdot(2\overrightarrow{a}+\overrightarrow{b})=17$.  
$(1)$ Find the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ and the value of $|\overrightarrow{a}+\overrightarrow{b}|$.  
$(2)$ Let $\overrightarrow{c}=m\overrightarrow{a}+2\overrightarrow{b}$, $\overrightarrow{d}=2\overrightarrow{a}-\overrightarrow{b}$. If $\overrightarrow{c}$ and $\overrightarrow{d}$ are collinear, find the value of the real number $m$.

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