How to prove the sequence $\{a_n\}$ is unbounded, which satisfies the recurrence relation $a_{n+1}=\ln |a_n|$?

When I browsed Zhihu(a Chinese Q&A community), I met this question. That is

Let $\{a_n\}$ be recursive s.t. $$a_1=2,\ a_{n+1}=\ln |a_n|(n\in \Bbb N).$$ Show that $\{a_n\}$ is unbounded.

I want to investigate a subsequence $\{a_{t_n}\}$ of $\{a_n\}$, where $t_n$ is greatest integer satisfying $$a_{t_n}=\min_{1\leqslant k\leqslant n}a_k.$$ Thus $a_{t_n}\to -A(<0),n\to \infty$.

However, it helps little with the origin question. So how can I solve it ?

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