
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  $n \geq 4$  be an even integer. Consider an  $n \times n$  grid. Two cells ( $1 \times 1$  squares) are *neighbors* if they share a side, are in opposite ends of a row, or are in opposite ends of a column. In this way, each cell in the grid has exactly four neighbors.
An integer from 1 to 4 is written inside each square according to the following rules:


- If a cell has a 2 written on it, then at least two of its neighbors contain a 1.
- If a cell has a 3 written on it, then at least three of its neighbors contain a 1.
- If a cell has a 4 written on it, then all of its neighbors contain a 1.


Among all arrangements satisfying these conditions, what is the maximum number that can be obtained by adding all of the numbers on the grid?

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