One of the reasons why the Blazing Inferno Sect also decided to join hands with Silver Arrows Sect was because they found out that the apprentice put a curse to one of Yu Yan's children.
To which that curse directly linked the fate of Yu Yan's child to some ancient prophecy.
But instead of the prophecy that talked about the destruction of the world, what the Blazing Inferno Sect has was the other version f the prophecy which was about salvation.
So aside from helping the Silver Arrows Sect to find their missing young mistress, the Blazing Inferno Sect also had another agenda.
They bestowed Yuan Tao with another special mission---and that was to make sure that one of Yu Yan's children should be sent inside the Magus Academia so the child would learn the way of magic.
It was for that same reason that Yuan Tao benevolently offered his help to Mu Lingxi back then.