Every possible length

Discrete Mathematics Level 4

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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