
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 two vectors $\overrightarrow{a},\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=1,|\overrightarrow{b}|=2$.  
$(1)$ If $({2\overrightarrow{a}-\overrightarrow{b}})\cdot({\overrightarrow{a}+\overrightarrow{b}})=-3$, find the angle $\theta$ between $\overrightarrow{a}$ and $\overrightarrow{b}$;  
$(2)$ If the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $60^{\circ}$, and the angle between the vectors $t\overrightarrow{a}+\overrightarrow{b}$ and $2\overrightarrow{a}+t\overrightarrow{b}$ is obtuse, find the range of the real number $t$.

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