When Animate Dead enters, it trigger, returning Brotherhood Outcast from your graveyard to the battlefield and attaching Animate Dead to it.
When Brotherhood Outcast enters, it triggers, targeting Dance of the Dead.
Holding priority, activate Umbral Collar Zealot by sacrificing Brotherhood Outcast.
Resolve the Brotherhood Outcast trigger, returning Dance of the Dead from your graveyard to the battlefield.
When Dance of the Dead enters, it trigger, returning Brotherhood Outcast from your graveyard to the battlefield and attaching Dance of the Dead to it.
When Brotherhood Outcast enters, it triggers, targeting Animate Dead.
Holding priority, activate Umbral Collar Zealot by sacrificing Brotherhood Outcast.
Resolve the Brotherhood Outcast trigger, returning Animate Dead from your graveyard to the battlefield.
Repeat from step 2.
This clever recursion loop relies on Brotherhood Outcast's enter-the-battlefield trigger to endlessly alternate retrieving Animate Dead and Dance of the Dead from your graveyard. As each Aura enters to reanimate Brotherhood Outcast, its mandatory leave-the-battlefield clause is neatly bypassed because you sacrifice the Outcast to Umbral Collar Zealot's activation in response, sending it right back to the graveyard to be fetched again. This self-sustaining engine lets you churn through your library with infinite surveil triggers while generating endless entry, death, and sacrifice events.