Yes, this solution works. Sadly, I came up with this solution after the
contest.

The solution that I crafted during the contest was the binary-search
method, but I had a little bug in deciding whether a given value would not
work. My method worked for all of the examples in the small test, and all
of the examples but 1 in my large test. It actually works for all of the
examples in the version of the large data set that can be downloaded now.
Very frustrating.

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