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

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