
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Let $\{a_n\}$ be a sequence whose terms are all positive numbers, and let $S_n$ be the sum of the first $n$ terms. For any positive integer $n$, $a_n$, $S_n$, and $a_n^2$ form an arithmetic sequence. Let $\{b_n\}$ be another sequence with $T_n$ as the sum of its first $n$ terms and $b_n = \frac{(\ln x)^n}{a_n^2}$. If for any real number $x \in (1, e]$ (where $e$ is the base of the natural logarithm) and any positive integer $n$, it always holds that $T_n < r (r \in \mathbb{N_+})$, then the minimum value of $r$ is __________.

Present the answer in LaTex format: \boxed{Your answer}
user