
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 ellipse E: $mx^2 + y^2 = 1$ ($m > 0$),
(I) If the right focus of the ellipse E has coordinates $(\sqrt{3}, 0)$, find the value of $m$;
(II) The triangle formed by any three distinct points on the ellipse E is called an inscribed triangle of the ellipse. If there are exactly three distinct isosceles right triangles inscribed in the ellipse E with $B(0, 1)$ as the right-angled vertex, find the range of $m$.

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