
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 $a_1=1$ and $n(a_{n+1}-a_n)=a_n+1$ for $n\in\mathbb{N}^*$, if there exists $t\in[1,3]$ such that the inequality $\frac{a_{n+1}}{n+1} \geq 2at-1$ holds for any positive integer $n$, find the maximum value of the integer $a$.

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