Tessellate S.T.E.M.S - Mathematics - School - Set 3 - Problem 2

$S = \dfrac{a(a+b)}{a^2+(a+b)^2} + \dfrac{b(b+c)}{b^2+(b+c)^2}+ \dfrac{c(c+d)}{c^2+(c+d)^2}+ \dfrac{d(d+a)}{d^2+(d+a)^2}$ $A = \bigg\{1, \dfrac{4}{5}, \dfrac{3}{7}, \dfrac{7}{5}, \dfrac{9}{4}, \dfrac{8}{5}, \dfrac{11}{7}, \dfrac{13}{5}, \dfrac{12}{7} \bigg\}$ How many elements $L \in A$ satisfy $S \leq L$ for all positive reals $a,b,c,d$ ?

This problem is a part of Tessellate S.T.E.M.S.