In coordinate space, there are two non-intersecting lines $L_{1}: \left\{\begin{array}{l} x = 1+t \\ y = 1-t \\ z = 2+t \end{array}\right.$, $t$ is a real number, $L_{2}: \left\{\begin{array}{l} x = 2+2s \\ y = 5+s \\ z = 6-s \end{array}\right.$, $s$ is a real number. Another line $L_{3}$ intersects both $L_{1}$ and $L_{2}$ and is perpendicular to both. If points $P$ and $Q$ are on $L_{1}$ and $L_{2}$ respectively and both are at distance 3 from $L_{3}$, then the distance between points $P$ and $Q$ is (17--1)$\sqrt{17\text{-}2}$. (Express as a simplified radical)
In coordinate space, there are two non-intersecting lines $L_{1}: \left\{\begin{array}{l} x = 1+t \\ y = 1-t \\ z = 2+t \end{array}\right.$, $t$ is a real number, $L_{2}: \left\{\begin{array}{l} x = 2+2s \\ y = 5+s \\ z = 6-s \end{array}\right.$, $s$ is a real number. Another line $L_{3}$ intersects both $L_{1}$ and $L_{2}$ and is perpendicular to both. If points $P$ and $Q$ are on $L_{1}$ and $L_{2}$ respectively and both are at distance 3 from $L_{3}$, then the distance between points $P$ and $Q$ is (17--1)$\sqrt{17\text{-}2}$. (Express as a simplified radical)