
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 any integer \( n \geq 2 \), let \( N(n) \) be the maximal number of triples \((a_i, b_i, c_i)\), \( i = 1, \ldots, N(n) \), consisting of nonnegative integers \( a_i, b_i \), and \( c_i \) such that the following two conditions are satisfied:
1. \( a_i + b_i + c_i = n \) for all \( i = 1, \ldots, N(n) \),
2. If \( i \neq j \), then \( a_i \neq a_j, b_i \neq b_j \), and \( c_i \neq c_j \).

Determine \( N(n) \) for all \( n \geq 2 \).

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