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Why install Oracle 126.96.36.199 on a different server.
Installing it on the same server in a different home would be the correct approach before you do the upgrade.
188.8.131.52 is released as a full release and not a patch release so you wouldn't have to patch the existing home.
in case the different server is requirement:
1. Install 184.108.40.206 on new servers as binary only, apply latest PSU if required. copy init.ora and other config files across
2. Run pre-upgrade scripts on 220.127.116.11 original server, failing the step will cause upgrade to fail
3. shutdown everything, mount your storage to new server
4. startup the instance in upgrade mode from 18.104.22.168 home
5. upgrade database as per docs: Complete Checklist for Manual Upgrades to 11gR2 [ID 837570.1]
6. update clients with new ip if listener config changed