CLL+1

CLL+1 is a last layer method proposed by Thom Barlow in 2011 and first developed by James Straughan in 2020. The method solves the four LL corners corners along with any single edge. The second step is L3E. The LL method can be used for when LL edges aren't always oriented and is called CLL+1. It can also be used when LL edges are always oriented, such as in ZZ or Petrus. This is called COLL+1.
History
In 2011 Thom Barlow posted example solves on the SpeedSolving Forums of a new LL method[1]. In this method, the four corners of the last layer were being solved along with one edge. Thom Barlow didn't yet have a system for how to solve an edge every time. He asked for ideas on how to make it work. A few ideas were provided[2][3], but none turned out to be viewed as viable. A few years later some had the idea of using an alternate algorithm based on the phased or unphased state of the edges for COLL to force the next step to be a UPerm EPLL case. In February 2017 Louis de Mendonça had the idea of applying the phased/unphased edges technique to methods where the LL edges aren't always oriented. The idea was to have one algorithm that orients the edges a specific way and another algorithm that orients the edges the opposite way of the first algorithm. This would be the same LL method as the original CLL+1 proposal by Thom Barlow. In 2020, James Straughan developed a system, called the Cycle Union System, for optimally solving groups of pieces[4]. He also independently developed the same edge orientation technique that was first thought of by Louis de Mendonça. By applying the Cycle Union System along with the edge orientation technique, development for CLL+1 and COLL+1 was completed in September 2020[5].
Pros
 High skip chance for the L3E step.
 No recognition time for L3E. Users will already know how their CLL+1 algorithm affects the remaining edges.
 L3E contains shorter algorithms compared to other second steps.
Cons
 A complete recognition system hasn't yet been developed.
Advancement
In its current form, the first step contains four algorithms per CLL case. Two of the algorithms cycle edges a certain way and the other two algorithms cycle the edges in a different way. Within each two algorithm set the two algorithms differ in the way that they affect the orientation of the edges. The Cycle Union System was applied for the edge permutation to reduce the number of cases for that aspect. However, the system wasn't applied for the edge orientation and, instead, algorithms with opposite edge orientations were used. If the Cycle Union System is applied to both edge permutation and edge orientation, it is possible that the number of cases for CLL+1 will be reduced.