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Open the Cisco AnyConnect Secure Mobility Client on the remote computer. In the text box, enter the server address:
vpn.utexas.edu
Click the "Connect" button.
A new window will open. In this window, enter 3 things:
Username: Your UT EID
Password: Your UT EID password
Duo Passcode: push
This will send a push notification to the Duo Mobile app on your mobile device. Open that app and tap the green button to allow the connection.
(Note that there are other options for the Duo Passcode field - you can retrieve a time-limited code directly from the Duo Mobile app, and enter it here. You can also enter "sms" (without quotes) instead of "push" in order to receive a text message for approval.)