
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 the unit vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is acute, and for any $(x,y)$ that satisfies $|x\overrightarrow{a}+y\overrightarrow{b}|=1$ and $xy\geqslant 0$, the inequality $|x+2y|\leqslant \frac{8}{\sqrt{15}}$ holds. Find the minimum value of $\overrightarrow{a}\cdot\overrightarrow{b}$.

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