Invariants much?

Imagine a circle divided into \(100\) sections (this is done by drawing lines through the center). The number \(1\) is placed in \(51\) of these sections, and in the rest the number \(0\) is placed.

In any move, you may add \(1\) to any two sections that share a side. In how many of these arrangements can you get the same number in every section?

HINT: An invariant is something that remains constant. There is one in particular that will be very useful!

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