from appian application , i will send some emails addresses to 3rd party application , that 3rd party app needs to verify those email addresses and sends the email addresses ids back .
do i need use webapi or integration for this scenarios
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If this is a synchronous call, an integration is what you want to use.
Simple: Use integrations to initiate calls from Appian to other systems and use Web APIs to initiate calls from other systems to Appian.