Let $$l_{1}, l_{2}, \ldots, l_{100}$$ be consecutive terms of an arithmetic progression with common difference $$d_{1}$$, and let $$w_{1}, w_{2}, \ldots, w_{100}$$ be consecutive terms of another arithmetic progression with common difference $$d_{2}$$, where $$d_{1} d_{2}=10$$. For each $$i=1,2, \ldots, 100$$, let $R_{i}$ be a rectangle with length $$l_{i}$$, width $$w_{i}$$ and area $A_{i}$. If $$A_{51}-A_{50}=1000$$, then the value of $$A_{100}-A_{90}$$ is __________.

The number of 4-digit integers in the closed interval [2022, 4482] formed by using the digits $$0,2,3,4,6,7$$ is _________.

Let $$A B C$$ be the triangle with $$A B=1, A C=3$$ and $$\angle B A C=\frac{\pi}{2}$$. If a circle of radius $$r>0$$ touches the sides $$A B, A C$$ and also touches internally the circumcircle of the triangle $$A B C$$, then the value of $$r$$ is __________ .

Consider the equation

$$ \int_{1}^{e} \frac{\left(\log _{\mathrm{e}} x\right)^{1 / 2}}{x\left(a-\left(\log _{\mathrm{e}} x\right)^{3 / 2}\right)^{2}} d x=1, \quad a \in(-\infty, 0) \cup(1, \infty) $$

Which of the following statements is/are TRUE?