Question 85
For a sequence of positive integers $x_1, x_2, \dots$, we perform the following operation: In the $i$-th step, we mark all rational numbers in the interval $[0, 1]$ with denominator $x_i$ (i.e., numbers of the form $j/x_i$ for $j \in \{0, 1, \dots, x_i\}$). Let $a_i$ be the length of the shortest interval whose two endpoints have been marked up to step $i$. Find all sequences $x_1, x_2, \dots$ such that $x_5 = 5$ and for every natural number $n$, we have:
$$a_1 + a_2 + \dots + a_n = 2 - \frac{1}{x_n}$$
$$a_1 + a_2 + \dots + a_n = 2 - \frac{1}{x_n}$$