Let \( A \) be a \( 3 \times 3 \) matrix such that \( A + A^{T} = O \). If
\[
A
\begin{bmatrix}
1 \\
-1 \\
0
\end{bmatrix}
=
\begin{bmatrix}
3 \\
3 \\
2
\end{bmatrix},
\quad
A^{2}
\begin{bmatrix}
1 \\
-1 \\
0
\end{bmatrix}
=
\begin{bmatrix}
-3 \\
19 \\
-24
\end{bmatrix}
\]
and
\[
\det\!\big(\operatorname{adj}(2\,\operatorname{adj}(A+I))\big)
= 2^{\alpha}\,3^{\beta}\,11^{\gamma},
\]
where \( \alpha, \beta, \gamma \) are non-negative integers, then the value of
\( \alpha + \beta + \gamma \) is ________.