Let $f:\mathbb{R}\to\mathbb{R}$ and $g:\mathbb{R}\to\mathbb{R}$ be two non-constant differentiable functions. If $f'(x)=(e^{(f(x)-g(x))})g'(x)$ for all $x\in\mathbb{R}$, and $f(1)=g(2)=1$, then which of the following statement (s) is (are) TRUE?

$f'(x)=e^{f(x)-g(x)}\,g'(x),\quad \forall x\in\mathbb{R}$

$f(1)=1,\qquad g(2)=1$

$\text{Compare } f(2),\ g(1) \text{ with } 1-\log_e 2$

$f'(x)e^{-f(x)}=g'(x)e^{-g(x)}$

$-\dfrac{d}{dx}\left(e^{-f(x)}\right)=-\dfrac{d}{dx}\left(e^{-g(x)}\right)$

$\Rightarrow \dfrac{d}{dx}\left(e^{-f(x)}-e^{-g(x)}\right)=0$

$\Rightarrow e^{-f(x)}-e^{-g(x)}=C\ (\text{constant})$

$x=1:\ C=e^{-1}-e^{-g(1)}$

$x=2:\ C=e^{-f(2)}-e^{-1}$

$e^{-1}-e^{-g(1)}=e^{-f(2)}-e^{-1}$

$\Rightarrow e^{-f(2)}+e^{-g(1)}=2e^{-1}$

$e^{-g(1)}>0 \Rightarrow e^{-f(2)}<2e^{-1}$

$\Rightarrow -f(2)<\ln 2-1 \Rightarrow f(2)>1-\log_e 2$

$\Rightarrow (2) \text{ true},\ (1) \text{ false}$

$e^{-f(2)}>0 \Rightarrow e^{-g(1)}<2e^{-1}$

$\Rightarrow g(1)>1-\log_e 2 \Rightarrow (3) \text{ true},\ (4) \text{ false}$

$\text{True statements: } (2) \text{ and } (3)$