A function $f: R \rightarrow R$ defined by

$$ f(x)=\left\{\begin{array}{c} 2 x+3, x \leq \frac{4}{3} \\ -3 x^2+8 x, x>\frac{4}{3} \end{array}\right. \text { is } $$

If $2^{4 n+3}+3^{3 n+1}$ is divisible by $P$ for all natural numbers $n$, then $P$ is

Consider the following statements

Statement $\mathrm{I} \cosh ^{-1} x=\tanh ^{-1} x$ has no solution

Statement II $\cosh ^{-1} x=\operatorname{coth}^{-1} x$ has only one solution

The correct answer is

The domain of the real valued function $f(x)=\log _{\sqrt{2}}\left(\sqrt{x^2+x}+\sqrt{x^2-x}\right)$ is