4. Given a hyperbola with asymptotes $y= \pm 2 x$, which passes through the intersection of the lines $x+y-3=0$ and $2 x-y+6=0$, then the length of the real axis of the hyperbola is $\qquad$

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Solving
\[
\left\{
\begin{array}{l}
x+y-3=0, \\
2 x-y+6=0
\end{array}
\Rightarrow
\left\{
\begin{array}{l}
x=-1, \\
y=4 .
\end{array}
\right.
\right.
\]
The hyperbola equation is \(\frac{y^{2}}{4}-x^{2}=\lambda\). Substituting the point \((-1,4)\) gives \(\lambda=3 \Rightarrow a^{2}=12 \Rightarrow 2 a=4 \sqrt{3}\).