  Non-Hermitian photonic systems with loss and gain attract much attention due
to their exceptional abilities in molding the flow of light. Introducing
asymmetry to the $\mathcal{PT}$-symmetric system with perfectly balanced loss
and gain, we reveal the mechanism of transition from the quasibound state in
the continuum (quasi-BIC) to the simultaneous coherent perfect absorption (CPA)
and lasing in a layered structure comprising epsilon-near-zero (ENZ) media. Two
types of asymmetry (geometric and non-Hermitian) are analyzed with the
scattering matrix technique. The effect of the CPA-lasing associated with the
quasi-BIC is characterized with the unusual linear dependence of the quality
factor on the inverse of asymmetry parameter. Moreover, the counter-intuitive
loss-induced-lasing-like behavior is found at the CPA-lasing point under the
non-Hermitian asymmetry. The reported features of non-Hermitian structures are
perspective for sensing and lasing applications.
