
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 given positive integer  $n$ , find the greatest positive integer  $k$ , such that there exist three sets of  $k$  non-negative distinct integers,  $A=\{x_1,x_2,\cdots,x_k\}, B=\{y_1,y_2,\cdots,y_k\}$  and  $C=\{z_1,z_2,\cdots,z_k\}$  with  $ x_j\plus{}y_j\plus{}z_j\equal{}n$  for any  $ 1\leq j\leq k$ .

<span style="font-size:85%"><span style="color:#0000FF">[Moderator edit: LaTeXified]</span></span>

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