Let the coefficient of \( x^{r} \) in the expansion of \( (x+3)^{n-1}+ (x+3)^{n-2}(x+2)+ (x+3)^{n-3}(x+2)^{2}+\cdots +(x+2)^{n-1} \) be \( \alpha_{\gamma} \). If \(\displaystyle \sum_{r=0}^{n}a_{\gamma}=\beta ^{n}-\gamma ^{n},\ \beta ,\gamma \in \mathbb{N} \), then the value of \( \beta ^{2}+\gamma ^{2} \) equals ________.
