  We develop an algorithm that combines the advantages of priority promotion -
one of the leading approaches to solving large parity games in practice - with
the quasi-polynomial time guarantees offered by Parys' algorithm. Hybridising
these algorithms sounds both natural and difficult, as they both generalise the
classic recursive algorithm in different ways that appear to be irreconcilable:
while the promotion transcends the call structure, the guarantees change on
each level. We show that an interface that respects both is not only effective,
but also efficient.
