  Based on the self-consistent equations of the order parameter $P_\infty$ and
the mean cluster size $S$, we develop a novel self-consistent simulation (SCS)
method for arbitrary percolation on the Bethe lattice (infinite homogeneous
Cayley tree). By applying SCS to the well-known percolation models, random bond
percolation and bootstrap percolation, we obtain prototype functions for
continuous and discontinuous phase transitions. By comparing the key functions
obtained from SCSs for the Achlioptas processes (APs) with a product rule and a
sum rule to the prototype functions, we show that the percolation transition of
AP models on the Bethe lattice is continuous regardless of details of growth
rules.
