I will be creating this in Jflap however, I am confused on this question. I have to create a DFSM in JFlap, but that's not what I$\text{'}$m stuck on$.$ I need help determining why the string is accepted or rejected. So it would be nice to show the DFSM and the reasoning behind it$.$

Note that

$$ Natural numbers mean non$-$negative integers.

$$ The transition in DFSM is a function, i$.$e$.,$ all the transitions need to be defined.

Accepted Strings: e $($empty$/$epsilon$),$ abb, bbbbb$,$ babbab, aababba

Rejected Strings: abba, aaaaaa, aabbabb