Two sequences $a=\{a_0, a_1, a_2,\ldots\}$ and $b=\{b_0,b_1,b_2,\ldots\}$ are defined as follows: \[a_0=0, ~a_1=1, ~a_n= a_{n-1} +b_{n-2} \hspace{2mm}\text{for}\hspace{2mm} n\ge2\]\[b_0=1, ~b_1=2, ~b_n=a_{n-2} +b_{n-1}\hspace{2mm}\text{for}\hspa ce{2mm} n\ge2\] what is the remainder when $a_{50}+b_{50}$ is divided by $5$?