
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  $n\ge 2$ , let  $N(n)$  be the maximum number of triples  $(a_j,b_j,c_j),j=1,2,3,\cdots ,N(n),$  consisting of non-negative integers  $a_j,b_j,c_j$  (not necessarily distinct) such that the following two conditions are satisfied:

(a)  $a_j+b_j+c_j=n,$  for all  $j=1,2,3,\cdots N(n)$ ;
(b)  $j\neq k$ , then  $a_j\neq a_k$ ,  $b_j\neq b_k$  and  $c_j\neq c_k$ .

Determine  $N(n)$  for all  $n\ge 2$ .

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