Let $f(x)$ be a function such that $f(1) = 1234$, $f(2)=1800$, and $f(x) = f(x-1) + 2f(x-2)-1$ for all integers $x$. Evaluate the number of divisors of
    \[\sum_{i=1}^{2022}f(i)\]

[i]2022 CCA Math Bonanza Tiebreaker Round #4[/i]