HMMT 2012 A1

Let $f$ be the function such that

$$f(x)= \begin{cases}2 x & \text { if } x \leq \frac{1}{2} \\ 2-2 x & \text { if } x>\frac{1}{2}\end{cases}$$

What is the total length of the graph of $\underbrace{f(f(\ldots f}_{2012 f^{\prime} s}(x) \ldots))$ from $x=0$ to $x=1$ ?