Call an integer $k$ debatable if the number of odd factors of $k$ is a power of two. What is the largest positive integer $n$ such that there exists $n$ consecutive debatable numbers? (Here, a power of two is defined to be any number of the form $2^m$, where $m$ is a nonnegative integer.)

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