
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  $f(x)=|2\{x\} -1|$  where  $\{x\}$  denotes the fractional part of  $x$ . The number  $n$  is the smallest positive integer such that the equation   $$ nf(xf(x)) = x $$   has at least  $2012$  real solutions  $x$ . Determine the value of  $n$ .

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