HMMT 2006 K10

Suppose $f$ and $g$ are differentiable functions such that

$$x g(f(x)) f^{\prime}(g(x)) g^{\prime}(x)=f(g(x)) g^{\prime}(f(x)) f^{\prime}(x)$$

for all real $x$. Moreover, $f$ is nonnegative and $g$ is positive. Furthermore,

$$\int_{0}^{a} f(g(x)) d x=1-\frac{e^{-2 a}}{2}$$

for all reals $a$. Given that $g(f(0))=1$, compute the value of $g(f(4))$.