Secret internet - we'd be crazy keeping record (4,3)

I believe the answer is:

deep web

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'secret internet' is the definition.
I know nothing about this answer so I can't judge whether this works.

'we'd be crazy keeping record' is the wordplay.
'crazy' indicates an anagram.
'keeping' indicates putting letters inside (some letters kept inside).
'record' becomes 'EP' (abbreviation for extended play record).
'wed'+'be'='wedbe'
'wedbe' is an anagram of 'deweb'.
'deweb' enclosing 'ep' is 'DEEP WEB'.

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I've seen this clue in The Independent.

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