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