Katie Ledecky and Michael Phelps each participate in 7 swimming events in the Olympics (and there is no event that they both participate in). Ledecky receives g_{L} gold, s_{L} silver, and b_{L} bronze medals, and Phelps receives g_{P} gold, s_{P} silver, and b_{P} bronze medals. Ledecky notices that she performed objectively better than Phelps: for all positive real numbers w_{b}<w_{s}<w_{g}, we have
w_{g} g_{L}+w_{s} s_{L}+w_{b} b_{L}>w_{g} g_{P}+w_{s} s_{P}+w_{b} b_{P}
Compute the number of possible 6-tuples \left(g_{L}, s_{L}, b_{L}, g_{P}, s_{P}, b_{P}\right).