
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 positive geometric sequence $\{a_n\}$ with a common ratio $q > 1$, and it satisfies $a_2=6$, $a_1a_3+2a_2a_4+a_3a_5=900$. Let the sum of the first $n$ terms of the sequence $\{a_n\}$ be $S_n$. If the inequality $\lambda a_n \leqslant 1+S_n$ holds for all $n \in \mathbb{N}^*$, then the maximum value of the real number $\lambda$ is ______.

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