Let \(729, 81, 9, 1, \ldots\) be a sequence and \(P_n\) denote the product of the first \(n\) terms of this sequence.
If
\[
2 \sum_{n=1}^{40} (P_n)^{\frac{1}{n}} = \frac{3^{\alpha} - 1}{3^{\beta}}
\]
and \(\gcd(\alpha, \beta) = 1\), then \(\alpha + \beta\) is equal to