Reply to "Comment on `Towards exact solutions of superconducting $T_c$ induced by electron-phonon interaction' "

In a series of papers, we have proposed a non-perturbative field-theoretic approach to deal with strong electron-phonon and strong Coulomb interactions. The key ingredient of such an approach is to determine the full fermion-boson vertex corrections by solving a number of self-consistent Ward-Takahashi identities. Palle (see Phys. Rev. B 110, 026501 (2024), arXiv:2404.02918) argued that our Ward-Takahashi identities failed to include some important additional terms and thus are incorrect. We agree that our Ward-Takahashi identities have ignored some potentially important contributions and here give some remarks on the role played by the additional terms.