Crossword Genius

See that on a broadcast and be amazed (3,4,3)

Ross

I believe the answer is:

eat ones hat

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'be amazed' is the definition.
I can't tell whether this definition defines the answer.

'see that on a broadcast' is the wordplay.
'broadcast' indicates an anagram (letters scattered or cast around).
'see'+'that'+'on'+'a'='seethatona'
'seethatona' anagrammed gives 'EAT ONE'S HAT'.

'and' is the link.

Can you help me to learn more?

(Other definitions for eat ones hat that I've seen before include "Offer to do this if one is proved wrong!" , "fulfil unlikely promise" , "Be very surprised (at unexpected outcome)" , "Perhaps do penance" .)

I've seen this clue in The Times.
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