
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
Let  $n$  be a positive integer not divisible by  $3$ . A triangular grid of length  $n$  is obtained by dissecting a regular triangle with length  $n$  into  $n^2$  unit regular triangles. There is an orange at each vertex of the grid, which sums up to 
\[\frac{(n+1)(n+2)}{2}\]
oranges. A triple of oranges  $A,B,C$  is *good* if each  $AB,AC$  is some side of some unit regular triangles, and  $\angle BAC = 120^{\circ}$ . Each time, Yen can take away a good triple of oranges from the grid. Determine the maximum number of oranges Yen can take.

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