Let the coefficient of $x^r$ in the expansion of $(x+3)^{n-1}+(x+3)^{n-2}(x+2)+(x+3)^{n-3}(x+2)^2+\ldots \ldots \ldots .+(x+2)^{n-1}$ be $\alpha_r$. If $\sum_{r=0}^n \alpha_r = \beta^n - \gamma^n, \quad \beta, \gamma \in \mathbb{N}$, then the value of $\beta^2+\gamma^2$ equals _________.