We can recognize the string aaabbb by the following sequence of moves:
(q0, aaabbb, 0)
(q1,
aabbb, 10)
(q1,
abbb, 110)
(q1,
bbb, 1110)
(q2,
bb, 110)
(q2,
b, 10)
(q2,
, 0)
(q3,
,
).
Since q3
F, the string is accepted.