
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 vectors $\overrightarrow{a}=(1,0),\overrightarrow{b}=(-1,2)$.  
$(1)$ If $|\overrightarrow{c}|=1$ and $\overrightarrow{c}∥(\overrightarrow{a}-\overrightarrow{b})$, find $\overrightarrow{c}$;  
$(2)$ If $2t\overrightarrow{a}-\overrightarrow{b}$ is perpendicular to $3\overrightarrow{a}+t\overrightarrow{b}$, find the value of the real number $t$.

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