
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 $\overrightarrow{a}=(1,0)$, $\overrightarrow{b}=(2,1)$,  
$(1)$ For what value of $k$ is $k \overrightarrow{a}- \overrightarrow{b}$ collinear with $\overrightarrow{a}+2 \overrightarrow{b}$.  
$(2)$ If $\overrightarrow{AB}=2 \overrightarrow{a}+3 \overrightarrow{b}$, $\overrightarrow{BC}= \overrightarrow{a}+m \overrightarrow{b}$, and points $A$, $B$, $C$ are collinear, find the value of $m$.

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