Let $x_1$, $x_2$, \ldots, $x_{10}$ be 10 numbers. Suppose that $x_i + 2x_{i+1} = 1$ for each $i$ from 1 through 9. What is the value of $x_1 + 512x_{10}$?

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