Temerant is a spherical planet with a radius of $1000$ kilometers. The government wants to build twelve towers of the same height on the equator of Temerant, so that every point on the equator can be seen from at least one tower. The minimum possible height of the towers can be written, in kilometers, as $a\sqrt{b} - c\sqrt{d} - e$ for positive integers $a, b, c, d$, and $e$ (with $b$ and $d$ not divisible by the square of any prime). Compute $a + b + c + d + e$.

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