Is there any way to know what counterfeit pass will work or is it all a crapshoot?
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I wasn't even aware any of the counterfeit passes could ever work; the way to get into Upper New Ark consistently is, after getting a counterfeit pass, also buy some pens at the hardware store, then tinker the two items together at certain spots (cassie's safehouse has a spot for this). If you succeed the stat check, you'll get a high quality counterfeit pass that always works, and if you fail, all that happens is you lose the two items (which, you would have lost the pass anyway if you tried to use it I think, and the pens cost almost nothing). Note that you will need to make a new pass every time you want to go up there though (at least right now).
Having pens and counterfeits on hand (I always keep around at least one spare of every consumable I can, just in case), going to one of the tinkering spots, and trying absolutely everything together to see what had results. Got some other good stuff this way (fairy dust in particular has several uses), but the high quality passes is by far the best use of these spots. Knew about Bearmarshal's Lin Lin solution too, and did that one the first time, but doing it yourself is far, far cheaper.