OBM 2018 Problem 4

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?