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Consider the cube shown below with its $8$ corners labelled $\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}, \mathrm{f}, \mathrm{g}$, and h . The figure is representative. All corners are to be colored such that any two corners that are connected by an edge must be of different colors. The minimum number of colors required to achieve this is $\_\_\_\_$

  1. $8$
  2. $4$
  3. $3$
  4. $2$

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