A real sequence $(a_n)_{n=0}^\infty$ is defined recursively by $a_0 = 2$ and the recursion formula $$ a_{n} = \begin{dcases} a_{n-1}^2 & \text{if $a_{n-1}<\sqrt3$} \\ \frac{a_{n-1}^2}{3} & \text{if $a_{n-1}\geq\sqrt 3$.} \end{dcases} $$Another real sequence $(b_n)_{n=1}^\infty$ is defined in terms of the first by the formula $$ b_{n} = \begin{dcases} 0 & \text{if $a_{n-1}<\sqrt3$} \\ \frac{1}{2^{n}} & \text{if $a_{n-1}\geq\sqrt 3$,} \end{dcases} $$valid for each $n\geq 1$. Prove that $$ b_1 + b_2 + \cdots + b_{2020} < \frac23. $$