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Geometry Level 5

\[\large\tan^{-1} x+\cos^{-1}\left(\dfrac{y}{\sqrt{1+y^2}}\right)=\sin^{-1} \dfrac{3}{\sqrt{10}} \]

\((x,y)\in\mathbb{Z^+}\)

The above equation has \(\color{blue}{\zeta}\) ordered \((x,y)\) solutions and let:

\[\large\left(x_i+y_i\right)_{\text{least}}\times \left(y_i-x_i\right)_{\text{max}}=\color{blue}{\beta}\]

where \((x_i,y_i)\) denote the \(ith\) ordered solution in the solution set of above equation.

\[\Large \color{blue}{\zeta}+\color{blue}{\beta}=\ ?\]

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