
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
Given \\(p\\): \\(\exists x \in (0, +\infty)\\), \\(x^2 - 2e\ln x \leqslant m\\); \\(q\\): The function \\(y = \left( \frac{1}{3} \right)^{2x^2 - mx + 2}\\) is monotonically decreasing on \\([2, +\infty)\\).
\\((1)\\) If \\(p \vee q\\) is a false statement, find the range of the real number \\(m\\);
\\((2)\\) If \\(p \vee q\\) is a true statement, and \\(p \wedge q\\) is a false statement, find the range of the real number \\(m\\).

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