
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\}\) have a sum of the first \\(n\\) terms denoted by \\(S_n\\), and it is known that \\(S_n = 2a_n - 2^{n+1} (n \in \mathbb{N}^*)\).
\\((1)\\) Find the general formula for the sequence \\(\{a_n\}\).
\\((2)\\) Let \\(b_n = \log_{\frac{a_n}{n+1}} 2\), and the sum of the first \\(n\\) terms of the sequence \\(\{b_n\}\) be \\(B_n\). If there exists an integer \\(m\\) such that for any \\(n \in \mathbb{N}^*\) and \\(n \geqslant 2\), \\(B_{3n} - B_n > \frac{m}{20}\) holds, find the maximum value of \\(m\\).

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