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

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