
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: The equation $x^2 - 2mx + 1 = 0$ has real roots, and the proposition q: The eccentricity $e$ of the hyperbola $\frac{y^2}{5} - \frac{x^2}{m} = 1$ lies in the interval $(1, 2)$. If the negation of q and the conjunction of p and q are both false propositions, find the range of the real number m.

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