
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
4. A positive integer n is given.Find the smallest  $k$  such that we can fill a  $3*k$  gird with non-negative integers 
such that: $\newline$   $i$ ) Sum of the numbers in  each column is  $n$ .
  $ii$ ) Each of the numbers  $0,1,\dots,n$  appears at least once in each row.

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