
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 $|\overrightarrow{b}|=2$, $\angle(\overrightarrow{a}, \overrightarrow{b}) = 30^{\circ}$, and for all $\lambda > 0$, $|\overrightarrow{a}-\lambda \overrightarrow{b}|\geqslant |\overrightarrow{a}- \overrightarrow{b}|$ always holds. Determine the value of $\overrightarrow{a}\cdot \overrightarrow{b}$.

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