Consider a matrix A = [α, β, γ; α^2, β^2, γ^2; β+γ, γ+α, α+β], where α, β, γ are three distinct natural numbers. If (det ( adj ( adj ( adj ( adj A…

Consider a matrix $\displaystyle \mathrm{A}=\left[\begin{array}{ccc}\alpha & \beta & \gamma \\ \alpha^2 & \beta^2 & \gamma^2 \\ \beta+\gamma & \gamma+\alpha & \alpha+\beta\end{array}\right]$, where $\displaystyle \alpha, \beta, \gamma$ are three distinct natural numbers.If $\displaystyle \frac{\operatorname{det}(\operatorname{adj}(\operatorname{adj}(\operatorname{adj}(\operatorname{adj} \mathrm{A}))))}{(\alpha-\beta)^{16}(\beta-\gamma)^{16}(\gamma-\alpha)^{16}}=2^{32} \times 3^{16}$, then the number of such $\displaystyle 3$ - tuples $\displaystyle (\alpha, \beta, \gamma)$ is \_\_\_\_.