
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 proposition $p: \exists x_0 \in [1,3]$, $x_0 - \ln x_0 < m$; proposition $q: \forall x \in \mathbb{R}$, $x^2 + 2 > m^2$.

(1) If $(\neg p) \wedge q$ is a true proposition, find the range of the real number $m$.

(2) If $p \vee q$ is a true proposition, and $p \wedge q$ is a false proposition, find the range of the real number $m$.

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