Let
\[\mathbf{M} = \begin{pmatrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ a & 2 & b \end{pmatrix}.\]If $\mathbf{M} \mathbf{M}^T = 9 \mathbf{I},$ then enter the ordered pair $(a,b).$

Note: For a matrix $\mathbf{A},$ $\mathbf{A}^T$ is the transpose of $\mathbf{A},$ which is generated by reflecting the matrix $\mathbf{A}$ over the main diagonal, going from the upper-left to the lower-right.  So here,
\[\mathbf{M}^T = \begin{pmatrix} 1 & 2 & a \\ 2 & 1 & 2 \\ 2 & -2 & b \end{pmatrix}.\]
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
Let
\[\mathbf{M} = \begin{pmatrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ a & 2 & b \end{pmatrix}.\]If $\mathbf{M} \mathbf{M}^T = 9 \mathbf{I},$ then enter the ordered pair $(a,b).$

Note: For a matrix $\mathbf{A},$ $\mathbf{A}^T$ is the transpose of $\mathbf{A},$ which is generated by reflecting the matrix $\mathbf{A}$ over the main diagonal, going from the upper-left to the lower-right.  So here,
\[\mathbf{M}^T = \begin{pmatrix} 1 & 2 & a \\ 2 & 1 & 2 \\ 2 & -2 & b \end{pmatrix}.\]
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
(-2,-1)
Let
\[\mathbf{M} = \begin{pmatrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ a & 2 & b \end{pmatrix}.\]If $\mathbf{M} \mathbf{M}^T = 9 \mathbf{I},$ then enter the ordered pair $(a,b).$

Note: For a matrix $\mathbf{A},$ $\mathbf{A}^T$ is the transpose of $\mathbf{A},$ which is generated by reflecting the matrix $\mathbf{A}$ over the main diagonal, going from the upper-left to the lower-right.  So here,
\[\mathbf{M}^T = \begin{pmatrix} 1 & 2 & a \\ 2 & 1 & 2 \\ 2 & -2 & b \end{pmatrix}.\]
train