point-by-point response to the main concerns raised by the reviewers. To Reviewer # 2

We thank all the reviewers for their insightful and encouraging comments. Per your suggestion, we will update the appendix by adding more explanations about the proof ideas. T, (a) and (b) hold for IFS. Similarly, we can extend convergence results in Theorem 4 in Appendix from FS to IFS. We expect that the approach developed in this paper will fuel this future investigation.