
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
An equilateral triangle with side length 3 is divided into nine smaller triangular cells, as shown in the picture. Initially, zeros are written in these cells. In one move, you can choose two numbers located in neighboring cells (sharing a side) and either add one to both or subtract one from both. Petya wants to make several moves so that afterward, the cells contain consecutive natural numbers \( n, n+1, \ldots, n+8 \) in some order. For which \( n \) can he achieve this?

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