HMMT 2024 A5

Compute the unique ordered pair $(x, y)$ of real numbers satisfying the system of equations

$$\frac{x}{\sqrt{x^{2}+y^{2}}}-\frac{1}{x}=7 \text { and } \frac{y}{\sqrt{x^{2}+y^{2}}}+\frac{1}{y}=4 .$$