Tiny Target III

You are shooting at $n \geq 2$ targets on the real line, say $T_1,\dots, T_n$. $T_i$ is of radius $a_i\varepsilon$ and is centered at $x_i$ for each $1 \leq i \leq n$. This means $T_1$ covers $[x_1 - a_1\varepsilon, x_1 + a_1\varepsilon]$, and similar for the rest. Assume that all the targets are disjoint and that $a_i\varepsilon <<< 1$ for all $i$. You have one dart to shoot at the line. Your goal is to maximize your probability of hitting one of the targets. You can choose where to center your throw on the line. If you choose to center your dart at $\mu$, the actual position that the dart lands on the real line is $X \sim N(\mu,\sigma^2)$, where $\sigma^2$ is fixed. Find the value of $\mu$ that maximizes the probability of hitting a target when $n = 10, x_i = a_i = i,$ and $\sigma^2 = 9$. If necessary, round your answer to the nearest thousandth.