If $\overrightarrow a,\:\overrightarrow b,\:\overrightarrow c$ are any three unit vectors such that $\overrightarrow a.\overrightarrow b=\overrightarrow a.\overrightarrow c=0$ and the angle between $\overrightarrow b\:\:and\:\:\overrightarrow c$ is $\large\frac{\pi}{6}$, then $\overrightarrow b\times\overrightarrow c=?$

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