
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 the statements $p$: $\exists x \in (0, +\infty)$, $x^2 - 2e\ln x \leq m$; and $q$: the function $y=(\frac{1}{3})^{2x^2-mx+2}$ is monotonically decreasing on $[2, +\infty)$.  
(1) If the statement $p \vee q$ is false, determine the range of the real number $m$.  
(2) If the statement $p \vee q$ is true, but $p \wedge q$ is false, determine the range of the real number $m$.

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