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Turing Machine But Way Worse, 419 bytes

0 0 0 1 1 0 0
1 0 1 1 A 0 0
0 1 0 1 2 0 0
1 1 1 1 a 0 0
0 2 0 1 3 0 0
1 2 1 1 b 0 0
0 3 0 1 4 0 0
1 3 1 1 c 0 0
0 4 0 1 5 0 0
1 4 1 1 d 0 0
0 5 0 1 6 0 0
1 5 1 1 e 0 0
0 6 0 1 7 0 0
1 6 1 1 f 0 0
0 7 0 1 7 0 1
1 7 1 1 7 0 1
0 A 0 1 a 0 0
1 A 1 1 a 0 0
0 a 0 1 b 0 0
1 a 1 1 b 0 0
0 b 0 1 c 0 0
1 b 1 1 c 0 0
0 c 0 1 d 0 0
1 c 1 1 d 0 0
0 d 0 1 e 0 0
1 d 1 1 e 0 0
0 e 0 1 f 0 0
1 e 1 1 f 0 0
0 f 0 1 0 1 0
1 f 1 1 0 1 0

Try it online!

Full cat program.