LRRL or RLLR

Must be symmetric, To all those who think so. LRRL or RLLR is a puzzle game in which you enjoy increasing your score by choosing left or right according to one recursive rule. You can enjoy it as long as you dont make the wrong choice. All choices are divided to the left or right when you start, and either contains all of your later choices. If you start from the left, consider the left-right-right-left as one left and complete the next left-right-right-left. Repeat this process. e.g. LRRL-RLLR-...