
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 maxima 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:

- $a_i+b_i+c_i=n$  for all  $i=1, \ldots, N(n)$ ,
- 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  $n\geq 2$ .

*Proposed by Dan Schwarz, Romania*

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