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 -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 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 the 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.