Rachel starts with the number $1000$ in her hands. Each minute, she flips a fair coin. If it lands heads, she adds 1 to the number by placing it in her left pocket, transforming it to $x+1$. If it lands tails, she inverts the number by placing it in her right pocket, transforming it to $x^{-1}$. After eight minutes, the expected value of the number in Rachel's hands is $E$. Compute $\left\lfloor\frac{E}{10}\right\rfloor$.

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