Given $\mathbf{A} = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix},$ there exist positive real numbers $x$ and $y$ such that
\[(x \mathbf{I} + y \mathbf{A})^2 = \mathbf{A}.\]Enter the ordered pair $(x,y).$
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
Given $\mathbf{A} = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix},$ there exist positive real numbers $x$ and $y$ such that
\[(x \mathbf{I} + y \mathbf{A})^2 = \mathbf{A}.\]Enter the ordered pair $(x,y).$
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
\left( \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}} \right)
Given $\mathbf{A} = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix},$ there exist positive real numbers $x$ and $y$ such that
\[(x \mathbf{I} + y \mathbf{A})^2 = \mathbf{A}.\]Enter the ordered pair $(x,y).$
train