
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
Let $\overrightarrow{a}=({2,0}),\overrightarrow{b}=({1,\sqrt{3}})$.  
$(1)$ Find the unit vector perpendicular to $\overrightarrow{a}+\overrightarrow{b}$.  
$(2)$ If the angle between the vectors $t\overrightarrow{a}+\overrightarrow{b}$ and $\overrightarrow{a}+t\overrightarrow{b}$ is obtuse, find the range of real number $t$.

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