Xintong plays a game of turning one six-digit number into another. The numbers can have leading zeros, but cannot go over 6 digits or below 0. He can only make the following moves, any number of times in any order:
R: rotate the last digit to the start, for example, 092347 \rightarrow 709234, or
A: add 1001 to the number, for example, 709234 \rightarrow 710235, or
S: subtract 1001 from the number, for example, 709234 \rightarrow 708233.
a. Show that it is possible to turn 202122 into 313233.
b. Show that turning 999999 into 000000 can be done in eight moves.
c. Show that any multiple of 11 remains a multiple of 11 after any sequence of moves.
d. Show that it is impossible to turn 112233 into 000000.