
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
In the Cartesian coordinate system, given vectors $\overrightarrow{a}=(1,1)$ and $\overrightarrow{b}=(2,-1)$.  
$(1)$ Find $|3\overrightarrow{a}-\overrightarrow{b}|$;  
$(2)$ If $\overrightarrow{m}=2\overrightarrow{a}-\overrightarrow{b}$ and $\overrightarrow{n}=t\overrightarrow{a}+\overrightarrow{b}$, and $\overrightarrow{m}\perp \overrightarrow{n}$, find the value of the real number $t$.

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