
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 an ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$), the two foci and the two vertices of the minor axis form a square, and its perimeter is $4\sqrt{2}$.
(1) Find the equation of ellipse $C$;
(2) Let line $l$ pass through point $B(0, m)$ ($m > 0$) and intersect ellipse $C$ at points $E$ and $F$. The point symmetric to $B$ with respect to the origin is $D$. If point $D$ always lies inside the circle with diameter $EF$, find the range of $m$.

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