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Let $\sum=\{0,1\}.$ Let $WW_{k}=\{ww|w\in \sum^{*}$ and $w$ is of length $k\}.$

1. Show that for each $k,$ no DFA can recognize $WW_{k}$ with fewer than $2^{k}$ states.
2. Describe a much smaller $NFA$ for $\overline{WW_{k}},$ the complement of $WW_{k}.$
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