
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 \( f(x) = |2\{x\} - 3| \) where \( \{x\} \) denotes the fractional part of \( x \). The number \( m \) is the smallest positive integer such that the equation \[mf(xf(x))=x\] has at least 100 real solutions. What is \( m \)?

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