Persevere as alert being taken under southern river? (4,4,2)

I believe the answer is:

stay with it

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'persevere' is the definition.
(I've seen this in another clue)

'alert being taken under southern river?' is the wordplay.
'alert being' becomes 'withit' (I am not sure about this - if you are sure you should believe this answer much more).
'taken under' is a charade indicator (letters next to each other).
'southern' becomes 's' (abbreviation).
'river?' becomes 'tay' (Scottish river).
's'+'tay'='stay'
'withit' after 'stay' is 'STAY WITH IT'.

'as' acts as a link.

Can you help me to learn more?

I've seen this clue in The Independent.

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