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To integrate Twilio's SMS functionality with Genesys Cloud, users need to connect their existing Twilio account to Genesys Cloud. This involves provisioning a Genesys webhook, setting up the integration in Genesys Cloud, importing Twilio phone numbers into Genesys Cloud, and configuring queue settings for inbound and outbound traffic. Additionally, users can leverage alpha senders, which allow for branded messaging with pre-registration required for certain countries. Users must also consider limitations on alpha sender usage, such as one-way messaging and opt-out requirements.