
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 that $m$ is a real constant. Proposition $p$: The equation $\frac{x^2}{2m} - \frac{y^2}{m-6} = 1$ represents an ellipse with foci on the $y$-axis; Proposition $q$: The equation $\frac{x^2}{m+1} + \frac{y^2}{m-1} = 1$ represents a hyperbola.

$(1)$ If proposition $p$ is true, find the range of values for $m$;

$(2)$ If proposition $q$ is false, find the range of values for $m$;

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