Define the binary operation $a \Delta b = ab + a - 1$. Compute the following expression:
\[
10 \Delta (10 \Delta (10 \Delta (10 \Delta (10 \Delta (10 \Delta (10 \Delta (10 \Delta (10 \Delta 10))))))))
\]
where $10$ is written $10$ times.

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