
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

Given a \( 2016 \times 2016 \) board, what is the smallest \( k \) such that the cells of the board can be colored with \( k \) colors while satisfying the following conditions:
1. One of the diagonals is colored in the first color.
2. Cells symmetric with respect to this diagonal are colored the same.
3. Any two cells located in the same row on different sides of the cell of the first color are colored differently (cells do not necessarily have to be adjacent to the cell of the first color).

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