
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 a sequence $\{a\_n\}$, where $a\_1=3$, $n(a_{n+1}-a_n)=a_n+1$, $n\in\mathbb{N}^*$. If for any $a\in[-1,1]$, $n\in\mathbb{N}^*$, the inequality $\frac{a_{n+1}}{n+1} < t^2-2at+1$ always holds, then the range of the real number $t$ is _______.

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