I want you to modify this turing machine so that it achieves the same result in a slightly different way. Do not attempt anything drastic as your solution must work:

createTuring({
    transitions: [
        { state: "start",  read: "a",   write: "A", move: "R", nextState: "get"    },
        { state: "start",  read: "b_",              move: "R", nextState: "reject" },
        { state: "get",    read: "a",   write: "_", move: "R", nextState: "findb"  },
        { state: "get",    read: "B",   write: "_", move: "R", nextState: "wipe"   },
        { state: "get",    read: "b",   write: "_", move: "R", nextState: "keep"   },
        { state: "findb",  read: "aB",              move: "R", nextState: "findb"  },
        { state: "findb",  read: "b",   write: "B", move: "L", nextState: "rewind" },
        { state: "findb",  read: "_",               move: "L", nextState: "reject" },
        { state: "rewind", read: "aB",              move: "L", nextState: "rewind" },
        { state: "rewind", read: "_",               move: "R", nextState: "get"    },
        { state: "wipe",   read: "aB",  write: "_", move: "R", nextState: "wipe"   },
        { state: "wipe",   read: "b",   write: "_", move: "R", nextState: "keep"   },
        { state: "wipe",   read: "_",               move: "L", nextState: "reject" },
        { state: "keep",   read: "a",               move: "R", nextState: "keep"   },
        { state: "keep",   read: "_",               move: "L", nextState: "pickup" },
        { state: "keep",   read: "b",   write: "_", move: "R", nextState: "trim"   },
        { state: "trim",   read: "ab",  write: "_", move: "R", nextState: "trim"   },
        { state: "trim",   read: "_",               move: "L", nextState: "pickup" },
        { state: "pickup", read: "a",   write: "_", move: "L", nextState: "roll"   },
        { state: "pickup", read: "_",               move: "L", nextState: "pickup" },
        { state: "pickup", read: "A",   write: "_",            nextState: "accept" },
        { state: "roll",   read: "a",               move: "L", nextState: "roll"   },
        { state: "roll",   read: "_",   write: "a", move: "R", nextState: "right"  },
        { state: "right",  read: "a",               move: "R", nextState: "right"  },
        { state: "right",  read: "_",               move: "L", nextState: "pickup" },
        { state: "roll",   read: "A",   write: "a",            nextState: "accept" },
    ],
    initState: "start",
    tape: "aaababaaaaabaa"
});

The number choice function
The number choice function is defined as follows.
Input: a sequence k, x1, x2, . . . , xn where
• k is a positive integer.
• For each i, xi
is a nonnegative integer.
• All these numbers are encoded in unary using repetition of the letter a , with the letter
b used as a separator (in effect, playing the role of the commas). So the input sequence
k, x1, x2, . . . , xn is encoded as the string
a
k
bax1 bax2 b · · · a
xn
Output: xk, i.e., the k-th of the x-numbers.
• This also is encoded in unary in the same way, but without any b at the end. So the
output xk is encoded as the string a
xk .
• Recall the definition of the output of a Turing machine from Lecture 18 slide 28 (or Course
Notes §18.8 p. 223). Once the machine enters the Accept state, it does not matter what
comes after the leftmost blank cell; those later cells are not considered to be part of the
output string.
Some example inputs and outputs:
input sequence input encoded as string output number output encoded as string
3,1,5,2 aaababaaaaabaa 2 aa
2,4,7,0,3 aabaaaabaaaaaaabbaaa 7 aaaaaaa
1,0,1 abba 0 ε [first tape cell must be empty]
1,3,2,0 abaaabaab 3 aaa
3,3,2,0 aaabaaabaab 0 ε [first tape cell must be empty]
0,1,5,2 babaaaaabaa undefined crash: invalid input, k = 0
4,1,5,2 aaaababaaaaabaa undefined crash: invalid input, k > n
Notes:
• Any input string that is not of the specified form should be rejected, by crashing. Examples
of situations that must lead to a crash include: k = 0; k > n.
• You can, and should, have extra letters in your tape alphabet that are not in the input alphabet
{a,b}.