
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
For each positive integer  $n$ , find the largest real number  $C_n$  with the following property. Given any  $n$  real-valued functions  $f_1(x), f_2(x), \cdots, f_n(x)$  defined on the closed interval  $0 \le x \le 1$ , one can find numbers  $x_1, x_2, \cdots x_n$ , such that  $0 \le x_i \le 1$  satisfying 
\[|f_1(x_1)+f_2(x_2)+\cdots f_n(x_n)-x_1x_2\cdots x_n| \ge C_n\]

*Marko Radovanović, Serbia*

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