DE-MO 2016 Problem 6

Let $$f(x_1,x_2,x_3,x_4,x_5,x_6,x_7)=x_1x_2x_4+x_2x_3x_5+x_3x_4x_6+x_4x_5x_7+x_5x_6x_1+x_6x_7x_2+x_7x_1x_3$$ be defined for non-negative real numbers $x_1,x_2,\dots,x_7$ with sum $1$ . Prove that $f(x_1,x_2,\dots,x_7)$ has a maximum value and find that value.