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We introduce a quasi-static particle-in-cell (PIC) code -- WAND-PIC -- which does not suffer from some of the common limitations of many quasi-static PICs, such as the need for a predictor-corrector method in solving electromagnetic fields. We derive the field equations under quasi-static (QS) approximation and find the explicit form of the "time" derivative of the transverse plasma current. After that, equations for the magnetic fields can be solved exactly without using the predictor-corrector method. Algorithm design and code structure are thus greatly simplified. With the help of explicit quasi-static equations and our adaptive step size, plasma bubbles driven by the large beam charges can be simulated efficiently without suffering from the numerical instabilities associated with the predictor-corrector method. In addition, WAND-PIC is able to simulate the sophisticated interactions between high-frequency laser fields and beam particles through the method of sub-cycling. Comparisons between the WAND-PIC and a first-principle full PIC code (VLPL) are presented. WAND-PIC is open-source, fully three-dimensional, and parallelized with the in-house multigrid solver. Scalability, time complexity, and parallelization efficiency up to thousands of cores are also discussed in this work.