4. Non-zero sequences $\left\{a_{n}\right\} 、\left\{b_{n}\right\}$ satisfy
$$
a_{n+1}=a_{n}+3 b_{n}, b_{n+1}=a_{n}+b_{n} \text {. }
$$

Let $x_{n}=\frac{a_{n}}{b_{n}}$. When $\left|x_{2000}+x_{2000}\right|$ takes the minimum value, the value of $x_{2000}$ is $\qquad$ .
$4.3 \pm \sqrt{2}$.
From $x_{n+1}=\frac{a_{n+1}}{b_{n+1}}=\frac{a_{n}+3 b_{n}}{a_{n}+b_{n}}=\frac{x_{n}+3}{x_{n}+1}=1+\frac{2}{x_{n}+1}$,
we know that $\dot{x}_{2006}=1+\frac{2}{x_{200}+1}$.
Therefore, $\left|x_{2005}+x_{2006}\right|=\left|1+x_{200}+\frac{2}{1+x_{200}}\right| \geqslant 2 \sqrt{2}$.
The equality holds if and only if $\left(1+x_{2000}\right)^{2}=2$.
Thus, $x_{2006}=1 \pm \sqrt{2}$, and we have
$$
x_{2007}=1+\frac{2}{1+x_{2006}}=1+\frac{2}{2 \pm \sqrt{2}}=3 \mp \sqrt{2} .
$$