A board measuring $64$ inches in length and $4$ inches in height is inclined such that the longer side of the board forms a $30^\circ$ angle with the ground. The distance from the ground to the highest point on the board can be expressed in the form $a + b\sqrt{c}$, where $a$, $b$, and $c$ are positive integers, and $c$ is not divisible by the square of any prime. What is the value of $a + b + c$?

Let's think step by step and output the final answer within \boxed{}.
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