
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
Given that when $x \gt 2$, the inequality $x\ln \left(x-1\right) \lt a\left(x-2\right)$ has a solution, the range of real number $a$ is ______; according to the previous inequality, when $a=2$, the inequality $\frac{1}{{3}^{2}\ln2}+\frac{1}{{4}^{2}\ln3}+\frac{1}{{5}^{2}\ln4}+\cdots+\frac{1}{(n+2)^{2}\ln(n+1)}＜t(n\in\mathbb{N}^{*})$ always holds, the minimum value of real number $t$ is ______.

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