Started with a ten-digit sum, originally, that’s spent (10)

I believe the answer is:

instigated

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'started' is the definition.
Although both the answer and definition are past participle verbs, I can't see how one could define the other.

'a ten-digit sum originally that's spent' is the wordplay.
'originally' suggests taking the first letters.
'that's spent' indicates an anagram.
The initial letter of 'sum' is 's'.
'a'+'ten'+'digit'+'s'='atendigits'
'atendigits' anagrammed gives 'INSTIGATED'.

'with' is the link.

Can you help me to learn more?

(Other definitions for instigated that I've seen before include "Got working" , "Stirred up" .)

I've seen this clue in The Times.

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