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P.S.
Haha. Looks like I missed the elephant: под-строить is really rare (and a little bit strange) in the meaning of "to build barn". It's first and common meaning is "to build up ground/base for...

Hmm... I feel that both of 'при-' and 'под-' can equally mean 'close to' and dictionaries really mention this word as synonyms.
In my perception subtle difference is derived from prefixes. "При-"...