Hoof, Paper, Scissors Triples timer 1s memory 1024MB

You have probably heard of the game "Rock, Paper, Scissors". The cows like to play a similar game they call "Hoof, Paper, Scissors".

The rules of "Hoof, Paper, Scissors" are simple. Two cows play against each other. They both count to three and then each simultaneously makes a gesture that represents either a hoof, a piece of paper, or a pair of scissors. Hoof beats scissors (since a hoof can smash a pair of scissors), scissors beats paper (since scissors can cut paper), and paper beats hoof (since the hoof can get a papercut). For example, if the first cow makes a "hoof" gesture and the second a "paper" gesture, then the second cow wins. Of course, it is also possible to tie, if both cows make the same gesture.

Now there are N (3≤N≤2⋅10^5) cows who want to play hoof paper scissors, and they each independently have a strategy of drawing from some fixed distribution. In particular, the ith cow's strategy is to play hoof, paper, or scissors with probabilities (\frac{h_i}{h_i+p_i+s_i},\frac{p_i}{h_i+p_i+s_i},\frac{s_i}{h_i+p_i+s_i}), respectively.

How many distinct triples of cows (A,B,C) are there such that A beats B on average, B beats C on average, and C beats A on average? We consider two triples the same if one equals the other up to a cyclic shift.