
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
Consider sets of real numbers $\left\{x_{1}, x_{2}, x_{3}, \ldots, x_{20}\right\}$, each between 0 and 1, such that $x_{1} x_{2} x_{3} \ldots x_{20}=\left(1-x_{1}\right)\left(1-x_{2}\right)\left(1-x_{3}\right) \ldots \left(1-x_{20}\right)$. Find the set for which the value of $x_{1} x_{2} x_{3} \ldots x_{20}$ is maximal.

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