<p>In this paper, we propose a novel Lagrange multiplier approach for both classical and nonlocal Allen-Cahn type equations that simultaneously satisfy the energy dissipation law and the maximum bound principle. By incorporating stabilizing terms, the first- and second-order numerical schemes we constructed fulfill energy stability and offer advantages in two key aspects. Firstly, our novel method obeys the original discrete energy dissipation law, which states that the discrete original energy associated with the numerical solutions decreases monotonically over time. Secondly, the proposed numerical schemes preserve the discrete maximum bound principle, meaning that if the initial condition is bounded by a specific constant, the maximum absolute value of the numerical solution remains bounded by this constant throughout the entire calculation process. Furthermore, we establish the existence of solutions to the resultant nonlinear algebraic equations that require solving within the numerical schemes. Finally, typical numerical experiments are presented to rigorously verify the accuracy and effectiveness of the proposed approach.</p>