Prove that (1 2 3 4) is not a product of 3-cycles

Prove that (1 2 3 4) is not a product of 3-cycles

Note that a 3-cycle (a b c) can be written as a product of two transpositions. For example,
(a b c) = (a b)(b c). Hence any 3-cycle is even. On the other hand, (1 2 3 4) = (1 2)(2 3)(3 4),
which shows it is odd. Any product of even permutations must be even, hence (1 2 3 4) cannot
be a product of 3-cycles