2022 Cayley Problem 24

A cube with edge length 8 is balanced on one of its vertices on a horizontal table such that the diagonal from this vertex through the interior of the cube to the farthest vertex is vertical. When the sun is directly above the top vertex, the shadow of the cube on the table is a regular hexagon. The area of this shadow can be written in the form a\sqrt{b}, where a and b are positive integers and b is not divisible by any perfect square larger than 1. What is the value of a + b?

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Full credit to this problem is given to the CEMC, you may view all Cayley contests here.