Let $\alpha$ and $\beta$ be real numbers such that $-\frac{\pi}{4}<\beta<0<\alpha<\frac{\pi}{4}$.

If $\sin (\alpha+\beta)=\frac{1}{3}$ and $\cos (\alpha-\beta)=\frac{2}{3}$, then the greatest integer less than or equal to

$$ \left(\frac{\sin \alpha}{\cos \beta}+\frac{\cos \beta}{\sin \alpha}+\frac{\cos \alpha}{\sin \beta}+\frac{\sin \beta}{\cos \alpha}\right)^{2} $$ is

If $y(x)$ is the solution of the differential equation

$$ x d y-\left(y^{2}-4 y\right) d x=0 \text { for } x > 0, y(1)=2, $$

and the slope of the curve $y=y(x)$ is never zero, then the value of $10 y(\sqrt{2})$ is

The greatest integer less than or equal to

$$ \int_{1}^{2} \log _{2}\left(x^{3}+1\right) d x+\int_{1}^{\log _{2} 9}\left(2^{x}-1\right)^{\frac{1}{3}} d x $$

is ___________.

The product of all positive real values of $x$ satisfying the equation

$$ x^{\left(16\left(\log _{5} x\right)^{3}-68 \log _{5} x\right)}=5^{-16} $$

is __________.