<p>In this paper, we focus on using a derivative-free projection method to solve the pseudo-monotone system of nonlinear equations. Specifically, we develop a new hybrid three-term search direction that incorporates a flexible conjugate parameter. This direction always satisfies the sufficient descent condition and the trust region property without the need for a line search. By integrating an inertial acceleration technique, an adaptive line search, and the hyperplane projection method, we propose an accelerated hybrid three-term conjugate gradient projection method. We establish the global convergence of our method without requiring the Lipschitz continuity of the mapping. Furthermore, under the assumption that the mapping is locally Lipschitz continuous, we provide a detailed analysis of both the asymptotic and non-asymptotic global convergence rates in terms of iteration complexity. Numerical experiments demonstrate the computational efficiency of our proposed method compared to four existing similar methods. Finally, we demonstrate the practical potential of our method by applying it to signal restoration and traffic assignment problems.</p>