
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
Non-zero vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $\overrightarrow{b}=(\sqrt{3},1)$, $〈\overrightarrow{a},\overrightarrow{b}〉=\frac{π}{3}$, $(\overrightarrow{a}-\overrightarrow{b})⊥\overrightarrow{a}$. Find the projection vector of vector $\overrightarrow{a}$ onto the direction of vector $\overrightarrow{b}$, and express the answer using $\overrightarrow{a}$ or $\overrightarrow{b}$.

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