Fractions and reciprocals

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For a positive integer f3b9b7a46559a729577c3ca04dd2d7e66de16b3f, consider the 4bdca93962a2beaffc172ca394e04fe05bd74ffd fractions0ae1534056a1dd5cb338fb6dbbde44d6d77fd2c1The product of these fractions equals 174fadd07fd54c9afe288e96558c92e0c1da733a, but if you reciprocate (i.e. turn upside down) some of the fractions, the product will change. Can you make the product equal 1? Find all values of 174fadd07fd54c9afe288e96558c92e0c1da733a for which this is possible and prove that you have found them all.

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