Factorial Product Is Square

Find the largest integer n200n \leq 200 such that for some positive integer k<nk< n the product n!k!n!k! is a perfect square.

Details and assumptions

The number n! n!, read as n factorial, is equal to the product of all positive integers less than or equal to nn. For example, 7!=7×6×5×4×3×2×1 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1.

If you think that no such nn exists, type your answer as 0.

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