
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
Esmeralda writes  $2n$  real numbers  $x_1, x_2, \dots , x_{2n}$ , all belonging to the interval  $[0, 1]$ , around a circle and multiplies all the pairs of numbers neighboring to each other, obtaining, in the counterclockwise direction, the products  $p_1 = x_1x_2$ ,  $p_2 = x_2x_3$ ,  $\dots$  ,  $p_{2n} = x_{2n}x_1$ . She adds the products with even indices and subtracts the products with odd indices. What is the maximum possible number Esmeralda can get?

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