a function in the unit ball of C(X) where X is a compact space is a limit of convex combinations of extreme points

Question

Suppose $f(x)$ is a function in $C(X)$ such that $\|f\|<1-\frac{2}{n}$. Then there exist n extreme points of the unit ball of $C(X)$, $g_1,\ldots,g_n$ such that $f(x)=\frac{1}{n}(g_1(x)+\cdots+g_n(x))$.

I know that $g$ is an extreme point of the unit ball of $C(X)$ if and only if $|g(x)|=1$ for all $x\in X$.