A Haiku is a Japanese poem of seventeen syllables, in three lines of five, seven, and five.

Take five good haikus. Scramble their lines randomly. What are the chances that you end up with five completely good haikus (with five, seven, five)?

Your answer will be \( \frac{m}{n} \) where \( m, n \) are numbers such that \( m, n \) are positive integers and \( \gcd(m, n) = 1 \).

Take this answer and add the numerator and denominator.

Let's think step by step and output the final answer within \boxed{}.
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