This study aims to develop a single integrated spoofing-aware speaker verification (SASV) embeddings that satisfy two aspects. First, rejecting non-target speakers' input as well as target speakers' spoofed inputs should be addressed. Second, competitive performance should be demonstrated compared to the fusion of automatic speaker verification (ASV) and countermeasure (CM) embeddings, which outperformed single embedding solutions by a large margin in the SASV2022 challenge. We analyze that the inferior performance of single SASV embeddings comes from insufficient amount of training data and distinct nature of ASV and CM tasks. To this end, we propose a novel framework that includes multi-stage training and a combination of loss functions. Copy synthesis, combined with several vocoders, is also exploited to address the lack of spoofed data. Experimental results show dramatic improvements, achieving an SASV-EER of 1.06% on the evaluation protocol of the SASV2022 challenge.