
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 \\(f(x)\\) be a function defined on \\(\mathbb{R}\\) that is not identically zero, and for any \\(a, b \in \mathbb{R}\\), it satisfies \\(f(a, b) = af(b) + bf(a)\\), \\(f(2) = 2\\). The sequence \\(\{a_n\}\\) satisfies \\(a_n = \frac{f(2^n)}{2^n} (n \in \mathbb{N}^*)\\).

\\((1)\\) Find the general formula for the sequence \\(\{a_n\}\\).

\\((2)\\) If there exists a positive integer \\(n \in [1, 10]\\) such that \\(ma_n^2 + 2a_n - 2m - 1 < 0\\) holds, find the range of the real number \\(m\\).

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