
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
There are three sets of balls, each set containing $n$ balls, with the balls in each set numbered $1, 2, \cdots, n$. The balls are colored using two colors: black and white, ensuring that balls with the same number are colored the same. If for any group of four numbers $a_{1}, a_{2}, a_{3}, a_{4}$ (which can be the same) such that $a_{1} + a_{2} + a_{3} = a_{4}$, there is at least one ball of each color among the three balls corresponding to $a_{1}, a_{2}, a_{3}$, then the largest possible value of $n$ is ___

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