Divide $a^{2^k} − b^{2^k}$ by $(a + b)(a ^2 + b ^2 )(a ^4 + b^ 4 )\cdots (a^ {2 ^{k-1}} + b^ {2 ^{k-1}} )$. If $a=1$ and $b=-1$, what is the result of the division?

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