
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
If \( m \) and \( n \) are both non-zero real numbers, and satisfy the conditions for the two sets
$$
\begin{array}{l}
A=\{z \mid |z + n \mathrm{i}| + |z - m \mathrm{i}| = n, z \in \mathbf{C}\}, \\
B=\{z \mid |z + n \mathrm{i}| - |z - m \mathrm{i}| = -m, z \in \mathbf{C}\}
\end{array}
$$
to be non-empty sets, then the range of values for \( \frac{n}{m} \) is \(\qquad\).

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