1990 AIME Problem 12

A regular 12-gon is inscribed in a circle of radius 12. The sum of the lengths of all sides and diagonals of the 12-gon can be written in the form

a + b \sqrt{2} + c \sqrt{3} + d \sqrt{6},

where a^{}_{}, b^{}_{}, c^{}_{}, and d^{}_{} are positive integers. Find a + b + c + d^{}_{}.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.