So I'm one of those people who didn't pay that much attention to ranking up factions over the years. However, now I'm faced with the problem of potentially not being able to complete my record book because of my rank 0 with New Monarchy -- I have rank 30 Dead Orbit and I'm almost at rank 25 Future War Cult. I need to know the fastest and most efficient way of leveling up my reputation with New Monarchy to rank 25 before the Age of Triumph t-shirt promotion ends.
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There's a video on YouTube by CoS Gaming (this is his updated AoT version linked above). He posted a video of doing a run around on plaguelands doing patrols. He shows which patrols to do, the order, and the best and worst case scenario for the patrols you'll randomly get. The nightfall buff is necessary first and foremost as mentioned in the other comment. But by doing his method of running patrols, I was able to get Dead Orbit up to rank 18 as of 4/29/17 and I just joined about 2 weeks ago and I have only played for about 4 hours this week and less than 10 last week. Most of it is RNG. But you will still be depending on strange coins for the weekends to get that added boost in your ranks. I'll guess between doing that method, regularly playing the game as you do, and bankrupting Xur by taking every piece of heavy ammo in existence, you can level up a faction in less than 2 weeks... [spoiler]FYI : Being a completionist sucks, I know because I'm trying to finish the book by August as well... [url=http://www.example.com][/url]but you only need about 70% of the nodes unlocked for the book to register than you did 100% / Rank 7 and get you the shirt...[/spoiler]