
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 vector $\overrightarrow{a}=(\frac{\sqrt{3}}{2},\frac{1}{2})$, $|\overrightarrow{b}|=2$, $|2\overrightarrow{a}-\overrightarrow{b}|=\sqrt{6}$, $\overrightarrow{a}⋅\overrightarrow{b}=$____; the coordinates of the projection vector of $\overrightarrow{b}$ onto $\overrightarrow{a}$ are ____.

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