
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 the sequence $\left\{a_n\right\}$ satisfies $a_1=1, a_{n+1}=\frac{a_n}{a_{n+2}}$ $(n\in\mathbb{N}^*)$, and $b_{n+1}=(n-\lambda)\left(\frac{1}{a_n}+1\right)$ $(n\in\mathbb{N}^*)$, $b_1=-\lambda$ and the sequence $\left\{b_n\right\}$ is monotonically increasing, determine the range of the real number $\lambda$.

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