I've done all the mars quests, have the hand cannon equipped and discarded the polaris lance quests at sttage 1/5 so as for it not to clash. I don't know why none are dropping doing public events, patrols or lost sectors
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You need to fully finish the Nascent Dawn 1/5 quest before you start earning stems. You may want to reobtain it from Ana just so you can complete it (it’s fairly short if I recall correctly).