Let \( A \) be a \( 3 \times 3 \) matrix such that \( X^{T} A X = 0 \) for all non-zero \( 3 \times 1 \) matrices 
, and \( \det(\operatorname{adj}(2(A+I))) = 2^{\alpha} 3^{\beta} 5^{\gamma} \), \( \alpha, \beta, \gamma \in \mathbb{N} \), then \( \alpha^{2} + \beta^{2} + \gamma^{2} \) is ________.
