"We will not tolerate this behaviour being directed at any of our players and will fully support them in dealing with such incidents," the South London club said on their website (www.cpfc.co.uk).
"Any action we can take unilaterally as a club will be taken immediately and we will also seek cooperation from other clubs where relevant, following that we will pursue the matter criminally with the full force of whatever the law allows."
Palace did not say what action they had taken or give further details.
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Zaha said earlier this year that he was subjected to racist abuse in "nearly every game".
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