
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  $f_{0}(x)=x$ , and for each  $n\geq 0$ , let  $f_{n+1}(x)=f_{n}(x^{2}(3-2x))$ . Find the smallest real number that is at least as large as
\[ \sum_{n=0}^{2017} f_{n}(a) + \sum_{n=0}^{2017} f_{n}(1-a)\]
for all  $a \in [0,1]$ .

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