Incremental count across 2 columns while considering a 3rd column

I have a list that tracking teams, specifically, the game number of the season. I also would like to start the count over every month. I have 3 columns to consider in the formula - 1) the month, 2) the home team, and 3) the visiting team. What I hope to accomplish is something like I have listed below. The # columns are blank in the real file - this is just a small example to illustrate what I'm after. Is something like this even possible? Attached is an excel file with this same data. I greatly appreciate any help and your expertise!

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Thank you so very much for your help! It works perfectly and it's very much appreciated!