
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 \( f(x) = |1 - 2x| \) for \( x \in [0, 1] \). Define \( f_1(x) = f(x) \), \( f_2(x) = f(f_1(x)) \), \( f_3(x) = f(f_2(x)) \), and generally \( f_{n+1}(x) = f(f_n(x)) \). Determine how many solutions the equation \( f_n(x) = \frac{1}{2} x \) has in the interval \([0, 1]\).

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