Enumerator for the language w#w^R? [closed]

I'm trying to build a Turing machine diagram for the language w#w^R, where w^R is the reverse of w, and w is a word made up of 0's and 1's. I'm trying to think of an algorithm but I can't think of something not extremely complicated that goes over all combinations of the alphabet for a given word length, then type # onto the print tape and copies it back in reverse.

Is there something I'm missing here? Could there be a simpler algorithm here?

Here is a simple Turing machine segment that reads a binary word $w$ starting at a position immediately to the right of the delimeter symbol # and replaces it with $w$'s successor in the lexicographic ordering. Here, L,R,S indicate moving the read/write head Left, Right, or Stay.
This simple and powerful (standard) trick avoids a brute-force enumeration by using the results of previous computations to produce future ones. If you're allowed multiple tapes, it's now quite easy to accomplish the task you defined. Otherwise, there are standard $k$-tape to $1$-tape conversions.
• The $\not{b}$ character represents a blank tape symbol. – Aryeh Mar 10 '17 at 8:24