Let $a_1, a_2, a_3, \dots$ be a sequence of numbers such that $a_{n+2} = 2a_n$ for all integers $n$. Suppose $a_1 = 1$, $a_2 = 3$,
\[ \sum_{n=1}^{2021} a_{2n} = c, \quad \text{and} \quad \sum\limits_{n=1}^{2021} a_{2n-1} = b. \] 
Then $c - b + \frac{c-b}{b}$ can be written in the form $x^y$, where $x$ and $y$ are integers such that $x$ is as small as possible. Find $x + y$.

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