
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 \( m \) be a positive real number such that for any positive real numbers \( x \) and \( y \), the three line segments with lengths \( x + y \), \( \sqrt{x^2 + y^2 + xy} \), and \( m \sqrt{xy} \) can form the three sides of a triangle. Determine the range of values for \( m \).

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