Prove if $\{x^iy^jz^k \mid i \le2j\text{ or }j \le 3k\}$ is regular or not $$L = \{x^iy^jz^k \mid i \le2j\text{ or }j \le 3k\}$$

To Prove: If given language is regular or not.

I know that it is not a regular language but I am not able to come up with the string which I can use in the pumping lemma to prove that it is not regular.

We can also divide $L$ into two parts:
$$\begin{align*}
L_1 &amp;= \{x^iy^jz^k \mid i \le 2j\}\\
L_2 &amp;= \{x^iy^jz^k \mid j \le 3k\}\,,
\end{align*}$$

so I just need the strings to be used in the pumping lemma for $L_1$ and $L_2$.
It's not regular. Hint: Let $p$ be the integer of the pumping lemma and pump the string $x^{6p}y^{3p}z^{2p}$.