
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 that the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\frac{{2π}}{3}$, $|{\overrightarrow{a}}|=1$, and $|{\overrightarrow{b}}|=2$.  
$(1)$ Find the value of $\overrightarrow{a}⋅\overrightarrow{b}$.  
$(2)$ If $2\overrightarrow{a}-\overrightarrow{b}$ is perpendicular to $t\overrightarrow{a}+\overrightarrow{b}$, find the value of the real number $t$.

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