Shuffles to restore the deck

A 5252-card deck is shuffled as follows: split into positions 11–2626 and 2727–5252, then interleave so the bottom half (2727–5252) takes the odd positions 1,3,…,511, 3, \dots, 51 and the top half (11–2626) takes the even positions 2,4,…,522, 4, \dots, 52. What is the minimum number of such shuffles needed to return the deck to its original order?

Show hints (2)+
  1. The shuffle is a fixed permutation; the answer is its order. This is an in-shuffle of 5252 cards.
  2. Its order is ord⁡53(2)\operatorname{ord}_{53}(2) (smallest kk with 2k≡1 mod 532^k\equiv1\bmod 53), which is 5252.

Answer

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52

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