  We are studying the $SU(3)$ gauge theory with 12 staggered fermions,
searching for the endpoint of the line of first-order phase transitions in the
mass--beta plane. This endpoint plays an important role in our understanding of
the phase diagram of this model. Having found this endpoint with high
statistics on a small lattice using unimproved staggered fermions, we are
working to find it on larger lattices and with improved actions. For an action
improved with nHYP-smeared staggered fermions, we discuss the effect of slowly
turning off the improvement on the broken shift symmetry phase.
