If the function \(f(x)=\) \(\left{\begin{matrix}\frac{2}{x}\left{\sin \left({k}_{1}+1\right)x+\sin \left({k}_{2}-1\right…

If the function \(f(x)=\) \(\left{\begin{matrix}\frac{2}{x}\left{\sin \left({k}_{1}+1\right)x+\sin \left({k}_{2}-1\right)x\right}, & x<0 \\ 4 & x=0 \\ \frac{2}{x}{\log }_{e}\left(\frac{2+{\mathrm{k}}_{1}\mathrm{x}}{2+{\mathrm{k}}_{2}\mathrm{x}}\right), & x>0\end{matrix}\right.\)

is continuous at \(x=0,\) then \({\mathrm{k}}_{1}^{2}+{\mathrm{k}}_{2}^{2}\) is equal to

[JEE Main 2025, 23 Jan (Shift 1)]