
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Let  $f$  be the function such that 
\[f(x)=\begin{cases}2x & \text{if }x\leq \frac{1}{2}2-2x & \text{if }x>\frac{1}{2}\end{cases}\]
What is the total length of the graph of  $\underbrace{f(f(\ldots f}_{2012\text{ }f's}(x)\ldots))$  from  $x=0$  to  $x=1?$ 

Present the answer in LaTex format: \boxed{Your answer}
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