
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 propositions $$p: \exists x_{0} \in \mathbb{R}, x_{0}^{2} + 2x_{0} - m - 1 < 0$$, and $$q: \text{for} \ \forall x \in [1, 4], x + \frac{4}{x} > m$$.
(1) Write the negation of proposition $p$; and find the range of real numbers $m$ when proposition $p$ is true;
(2) If $p$ and $q$ are true and false respectively, find the range of the real number $m$.

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