
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
Let $p: \forall x \in \mathbb{R}, x^{2} + 2m - 3 > 0$ and $q: \exists x \in \mathbb{R}, x^{2} - 2mx + m + 2 < 0$.  
$(1)$ If proposition $p$ is a true statement, find the range of real number $m$;  
$(2)$ If proposition $q$ is a true statement, find the range of real number $m$;  
$(3)$ If at least one of the propositions $p$ or $q$ is true, find the range of real number $m$.

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