
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
Let $\{a_n\}$ be a sequence of positive numbers such that for any $n \in \mathbb{N}^*$, we have $a_1^3 + a_2^3 + a_3^3 + \dots + a_n^3 = S_n^2$, where $S_n$ denotes the sum of the first $n$ terms of the sequence $\{a_n\}$.

(1) Find the general formula for the sequence $\{a_n\}$;
(2) Let $b_n = 3^n + (-1)^{n-1}\lambda \cdot 2^{a_n}$ (where $\lambda$ is a non-zero constant, $n \in \mathbb{N}^*$), determine whether there exists an integer $\lambda$, such that for any $n \in \mathbb{N}^*$, we have $b_{n+1} > b_n$.

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