
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
In the sequence $\{a_n\}$, we have $a_1=0$, and $a_n-a_{n-1}-1=2(n-1)$ for $(n\in\mathbb{N}^*, n\geqslant 2)$. If the sequence $\{b_n\}$ satisfies $b_n=n\cdot \sqrt{a_{n+1}+1}\cdot \left( \frac{8}{11}\right)^{n-1}$, then the maximum term of the sequence $\{b_n\}$ is the $\boxed{6}$-th term.

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