
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 $\overrightarrow{a}=(x,1,0)$, $\overrightarrow{b}=(-1,y,2)$, $\overrightarrow{c}=(2,-2,1)$, $|\overrightarrow{b}|=\sqrt{5}$, $\overrightarrow{a}⊥\overrightarrow{c}$,  
$(1)$ If $\overrightarrow{a}+k\overrightarrow{b}$ and $2\overrightarrow{a}+\overrightarrow{b}$ are collinear, find the real number $k$;  
$(2)$ If the angle between the vectors $\overrightarrow{a}+k\overrightarrow{b}$ and $2\overrightarrow{a}+\overrightarrow{b}$ is acute, find the range of the real number $k$.

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