Use Mathematica to split a sequence according to other two sequences

I want to use Mathematica to solve the following problem:

For example I have a sequence 101. I want to compare it with $ 1101$ and $ 0101$ . The comparison has the following procedure:

  1. Check the first term of $ 101$ is $ 1$ or $ 0$ . If it is $ 1$ , compare $ 101$ to $ 1101$ , term by term; If it is $ 0$ , compare $ 101$ to $ 0101$ . Stop the process before the first term they are different, or all the terms of $ 101$ have been compared without stopping the process, and report all the terms that have been compared.

In our case, the first term of $ 101$ is $ 1$ , so we compare it with $ 1101$ . Then, $ 101$ and $ 1101$ only has one term in common, the first term $ 1$ . So the program should report $ 1$ , and go to next step.

  1. Recording the remaining sequence of $ 101$ .

In our case, as only $ 1$ is reported, the remaining sequence is $ 01$ .

  1. Restart the process. Check the first term of $ 01$ is $ 0$ or $ 1$ . If it $ 0$ , compare $ 01$ with $ 0101$ , if it is $ 1$ , compare $ 01$ with $ 1101$ . Stop the process before the first term they are different, or the sequence of $ 01$ has been run out. Report all the timers that have been compared.

*In our case, the first term of $ 01$ is $ 0$ , so we compare $ 01$ with $ 0101$ . Then the first two terms agree, and then $ 01$ ran out. The program should report $ 01$ , and then stop.

  1. Repeating the process again and again until there is no remaining sequence.

I tried to use the commend "If" to write this but it did not work, since I did not know how to let Mathematica to "remember" what has been compared.

Then, I tried to use commend "Order" and "Sort", but it seems that I need to program a comparison function.

Is there anyway for me to achieve this using Mathematica? Thank you!