
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 $y=x^{2}-2mx+1$ has two distinct roots.  
$(1)$ 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}
user