Let $f:[0, \infty) \rightarrow R$ be a continuous function such that
$$
f(x)=1-2 x+\int_0^x e^{x-1} f(t) d t
$$
for all $\mathrm{x} \in[0, \infty)$. Then, which of the following statement(s) is (are) TRUE ?
Select ALL correct options:
A
The curve $y=f(x)$ passes through the point $(1,2)$
B
The curve $y=f(x)$ passes through the point $(2,-1)$
C
The area of the region $\left\{(x, y) \in[0,1] \times R: f(x) \leq y \leq \sqrt{1-x^2}\right\}$ is $\frac{\pi-2}{4}$
D
The area of the region $\left\{(x, y) \in[0,1] \times R: f(x) \leq y \leq \sqrt{1-x^2}\right\}$ is $\frac{\pi-1}{4}$
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