Excitedly heading for Rome and Italy, beaming? (9)

I believe the answer is:

radiantly

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'excitedly heading for rome and italy beaming?' is the definition.
I can't judge whether this definition defines the answer.

'excitedly heading for rome and italy' is the wordplay.
'excitedly' indicates an anagram (letters excited or stimulated into a new form).
'heading for' suggests taking the first letters.
The first letter of 'rome' is 'r'.
'r'+'and'+'italy'='randitaly'
'randitaly' with letters rearranged gives 'RADIANTLY'.

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I've seen this clue in The Independent.

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