
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 the sequence \\( \{a_n\} \\) consist of positive numbers, and for any \\( n \in \mathbb{N}^* \\), the following holds: \\( a_1^3 + a_2^3 + a_3^3 + \cdots + a_n^3 = S_n^2 \\), where \\( S_n \\) is 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 nonzero constant, \\( n \in \mathbb{N}^* \\). Is there an integer \\( \lambda \\) such that for any \\( n \in \mathbb{N}^* \\), \\( b_{n+1} > b_n \\) holds? If such a \\( \lambda \\) exists, determine its value; if not, explain the reason.

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