If $\cosh 2 x=199$, then $\cot h x=$

If $a=\operatorname{Im}\left(\frac{1+z^2}{2 i z}\right)$ and $z$ is any non-zero complex number such that $|z|=1$, then $a=$

If $(3+4 i)^{2025}=5^{2023}(x+i y)$, then $\sqrt{x^2+y^2}=$

If $\left(\frac{\cos \theta+i \sin \theta}{\sin \theta+i \cos \theta}\right)^{2024}+\left(\frac{1+\cos \theta+i \sin \theta}{1-\cos \theta+i \sin \theta}\right)^{2025}=x+i y$ then the value of $x+y$ at $\theta=\frac{\pi}{2}$ is