HMMT 2020 T1

Let $n$ be a positive integer. Define a sequence by $a_{0}=1, a_{2 i+1}=a_{i}$, and $a_{2 i+2}=a_{i}+a_{i+1}$ for each $i \geq 0$. Determine, with proof, the value of $a_{0}+a_{1}+a_{2}+\cdots+a_{2^{n}-1}$.