The Lucas numbers $L_n$ is defined as $L_0=2$, $L_1=1$, and $L_n=L_{n-1}+L_{n-2}$ for $n\ge 2$. Let $r=0.21347\dots$, whose digits are Lucas numbers. When numbers are multiple digits, they will "overlap", so $r=0.2134830\dots$, NOT $0.213471118\dots$. Express $r$ as a rational number $\frac{q}{p}$ where $p$ and $q$ are relatively prime.