
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
On a $123 \times 123$ board, each cell is painted either purple or blue according to the following conditions:

- Each purple cell that is not on the edge of the board has exactly 5 blue cells among its 8 neighbors.
- Each blue cell that is not on the edge of the board has exactly 4 purple cells among its 8 neighbors.

Note: Two cells are neighbors if they share a side or a vertex.

(a) Consider a $3 \times 3$ sub-board within the $123 \times 123$ board. How many cells of each color can there be in this $3 \times 3$ sub-board?

(b) Calculate the number of purple cells on the $123 \times 123$ board.

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