# Every possible length

Let $$S$$ be a subset of $$[0,1]$$ consisting of a union of $$10$$ disjoint closed intervals $$I_1, I_2, \ldots, I_{10}.$$

Suppose $$S$$ has the property that for every $$d \in [0,1],$$ there are two points $$x,y \in S$$ such that $$|x-y|=d.$$

Letting $$s = \sum\limits_{n=1}^{10} \text{length}(I_n),$$ what is the minimum possible value of $$s?$$

Your answer should be a rational number $$\frac{p}{q},$$ where $$p$$ and $$q$$ are coprime positive integers.

Find $$p+q.$$

Bonus: Describe the sets $$S$$ for which the minimum is attained.

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