I feel really stupid asking about this, but there is so much terminology and stuff to learn, that I'm really not sure what this is (a bud, or what??):

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It's that green thing that's growing in between two blooms that are standing back to back so to speak.

Also, as you can see in this photo, one of the flowers is really drooping and looking like it's going to fall off soon. Should I just let it do its thing, or am I supposed to pinch it off?

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Thanks for putting up with my dumb questions. Hopefully, some day I'll have enough knowledge to help another newbie.