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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