Let $f(x) = x^3 + 3x - 1$ have roots $a, b, c$. Given that \(\frac{1}{a^3 + b^3} + \frac{1}{b^3 + c^3} + \frac{1}{c^3 + a^3}\) can be written as \(\frac{m}{n}\), where \(m, n\) are positive integers and \(\gcd(m, n) = 1\), find \(100m + n\).

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