
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 $m\in \mathbb{R}$. Proposition $p: \forall x \gt -2$, $x+\frac{49}{x+2}≥6\sqrt{2}m$; proposition $q: \exists x_{0}\in \mathbb{R}$, ${x}_{0}^{2}-m{x}_{0}+1=0$.  
$(1)$ If proposition $p$ is true, find the maximum value of $m$;  
$(2)$ If one proposition is true and the other is false, find the range of values of $m$.

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