The only difference between easy and hard versions is that you should complete all the projects in easy version but this is not necessary in hard version.

Polycarp is a very famous freelancer. His current rating is $$$r$$$ units.

Some very rich customers asked him to complete some projects for their companies. To complete the $$$i$$$-th project, Polycarp needs to have at least $$$a_i$$$ units of rating; after he completes this project, his rating will change by $$$b_i$$$ (his rating will increase or decrease by $$$b_i$$$) ($$$b_i$$$ can be positive or negative). Polycarp's rating should not fall below zero because then people won't trust such a low rated freelancer.

Polycarp can choose the order in which he completes projects. Furthermore, he can even skip some projects altogether.

To gain more experience (and money, of course) Polycarp wants to choose the subset of projects having maximum possible size and the order in which he will complete them, so he has enough rating before starting each project, and has non-negative rating after completing each project.

Your task is to calculate the maximum possible size of such subset of projects.