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

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