
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
For a positive integer \( k \), let the set
\[ 
R_{n} = 
\begin{cases} 
\{-k, -(k-1), \cdots, -1, 1, \cdots, k-1, k\}, & \text{if } n = 2k; \\ 
\{-k, -(k-1), \cdots, -1, 0, 1, \cdots, k-1, k\}, & \text{if } n = 2k+1.
\end{cases} 
\]

A device contains some balls and some red or white strings connecting pairs of balls. A "marking" is a coloring of each ball with one of the elements in \( R_n \). A marking is called "good" if and only if any two balls connected by a string have different colors; a marking is called "insightful" if and only if any two balls connected by a white string have different colors and any two balls connected by a red string have colors whose sum is not zero.

For a fixed positive integer \( n \geq 3 \), assume that any device has a "good" marking regarding \( R_n \), and it also has an "insightful" marking regarding \( R_m \). Find the smallest positive integer \( m = m(n) \).

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