
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) = \log_{\frac{1}{2}} \frac{1-ax}{x-1} + x\\) be an odd function, where \\(a\\) is a constant.

\\((1)\\) Find the value of \\(a\\);

\\((2)\\) Determine the monotonicity of the function \\(f(x)\\) on the interval \\(x \in (1, +\infty)\\) and explain the reason;

\\((3)\\) If for every \\(x\\) value in the interval \\([3,4]\\), the inequality \\(f(x) > \left( \frac{1}{2}\right)^x + m\\) always holds, find the range of the real number \\(m\\).

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