
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 functions \( f_{n}(x) \) (for \( n = 1, 2, 3, \ldots \)) are defined as follows:

\[ 
f_{1}(x) = 4(x - x^{2}) \quad (0 \leqslant x \leqslant 1) 
\]

\[ 
f_{n+1}(x) = f_{n}( f_{1}(x) ) \quad (n = 1, 2, \ldots) 
\]

Let \( a_{n} \) denote the number of values of \( x \) in \([0,1]\) that make \( f_{n}(x) \) attain its maximum value, and \( b_{n} \) denote the number of values of \( x \) in \([0,1]\) that make \( f_{n}(x) \) attain its minimum value. Find the expressions for \( a_{n} \) and \( b_{n} \) in terms of \( n \).

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