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P.S.
Maybe there is some non-restrict rule: "пода(ва)ть" is preferable if object (slightly) goes up and down during process of transfer. For example "подать нищему (милостыню)" (get coins and drop...

No. Hm... It is difficut for me to express meaning of "по-" in this case... Maybe it is case of memorizing.
There are some rules here: https://ru.wiktionary.org/wiki/%D0%BF%D0%BE- , and (3) is...