
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 \( a_{1} = 1 \) and \( a_{n+1} = \lambda a_{n}^{2} + 2 \left(n \in \mathbf{N}^{*}\right) \), if the sequence \(\{a_{n}\}\) is bounded, i.e., there exists a constant \( M \) such that \( a_{n} \leq M \) for all \( n \in \mathbf{N}^{*} \), find the maximum value of the real number \( \lambda \).

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