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Remorseful
Genuine Regret
Use 'remorseful' when the person feels sincere regret and wants to correct their mistake. 
Apology Included
'Remorseful' often accompanies an apology since it conveys the feeling of wanting to make amends. 
Not Just Sadness
'Remorseful' is deeper than just feeling sad; it involves acknowledging wrongdoing and feeling morally compelled to rectify it. 