
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

The function \( f_{n}(x) \) for \( n=1, 2, 3, \ldots \) is defined as follows: \( f_{1}(x) = 4 \left( x - x^2 \right) \) for \( 0 \leq x \leq 1 \), and \( f_{n+1}(x) = f_{n} \left( f_{1}(x) \right) \) for \( n = 1, 2, \ldots \).

Let \( a_{n} \) be the number of values of \( x \) in \([0,1]\) where \( f_{n}(x) \) attains its maximum, and let \( b_{n} \) be the number of values of \( x \) in \([0,1]\) where \( f_{n}(x) \) attains its minimum. Express \( a_{n} \) and \( b_{n} \) in terms of \( n \).

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