Pumping Lemma for irregular language, how to choose and divide the string The language on the alphabet $\{a,b\}$ is $$L=\{ab^nab^{n+1} |n&gt;0\}$$

so i have to take a string w = xyz:

what's the best string to choose such that $|w| &gt; n$? and then what about the division?

like $ w=ab^nab^{n+1}$ is a correct option? Then how can I divide this into three pieces? My attempt:

$ x = ab^s $ with  $ 0 ≤ s ≤ n -1 $

$ y= b^ta $   with   $  0 &lt; t ≤ n - t $

$ z = b^k $   with        $ 0 ≤ k &lt; n - t - s $
w = abn-1abn.

We have to consider all the division xyz = w with |y| >= 1 and |xy| &lt;= n. So

x = apbr

y = aqbs

z = btabn

where:


p + q = 1
r > 0 implies p = 1
q + s > 0
r + s + t = n - 1


Let's now consider w' = xy0z=apbr+tabn. If x is the empty string, then w' is missing the initial 'a' (and maybe some b), so w' is not in L. Else if x is not empty, we know that y contains at least one occurrence of 'b', so w' can't be in the form abnabn+1 and w' is not in L.