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 
\[ \sum_{i=1}^{2022} f(i) \]

Let's think step by step and output the final answer within \boxed{}.
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