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Actually, the number of possible positions has been estimated to ~5.1044, less than the ~1080 atoms in the universe (but that's indeed still very large).
There are engines and websites that can do that, like https://syzygy-tables.info/. The draw in game 6 was a tightrope walk into a 50 move repeat or perpetual check or 3-fold repeat, depending on how white played, and often required black to play one of two or three moves to hold the draw.
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