Problem 175 Algae Robot

This seems to be a difficult problem to solve. My approach is to find some periodicity in the movement of the robot that can be extrapolated to the large numbers given in the problem. The robot starts at x=0,y=0,looking toward +y. Using this as a reference point,I find that the next 100 times that the robot is in this state are at these positions on the string:

10,20,178,188,198,356,366,376,3194,3204,3214,3372,3382,3392,3550,3560,3570,6388,6398,6408,6566,6576,6586,6744,6754,6764, 57314,57324,57334,57492,57502,57512,57670,57680,57690,60508,60518,60528,60686,60696,60706,60864,60874,60884,63702,63712, 63722,63880,63890,63900,64058,64068,64078,114628,114638,114648,114806,114816,114826,114984,114994,115004,117822,117832, 117842,118000,118010,118020,118178,118188,118198,121016,121026,121036,121194,121204,121214,121372,121382,121392,1028458, 1028468,1028478,1028636,1028646,1028656,1028814,1028824,1028834,1031652,1031662,1031672,1031830,1031840,1031850,1032008, 1032018,1032028,1034846,1034856

There seems to be a vague pattern here, but with large unpredictable gaps, so I wouldn't call it periodic. I expect that choosing any other reference point woul yield similar results.

That was cryptic enough, but I'll take it under advisement.