
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  $f(x)=2x(1-x), x\in\mathbb{R}$  and denote  $f_n=f\circ f\circ ... \circ f$ ,  $n$  times.
(a) Find  $\lim_{n\rightarrow\infty} \int^1_0 f_n(x)dx$ .
(b) Now compute  $\int^1_0 f_n(x)dx$ .

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