Do a stretch up the river (5)

I believe the answer is:
reach
'do a stretch' is the definition.
'reach' can be an answer for 'do' (thesaurus). I am not sure about the 'a stretch' bit.
'up the river' is the wordplay.
I cannot quite understand how this works, but
'up' could be 'each' (I've seen this in another clue) and 'each' is found in the answer.
'river' could be 'r' (abbreviation) and 'r' is found in the answer.
No letters remain.
This explanation may well be incorrect...
Can you help me to learn more?
(Other definitions for reach that I've seen before include "Extend an arm" , "Be received" , "It's an advantage for a boxer" , "Put out the hand for a part of the river" , "Get to < throw up in the old" .)