
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 an $n \times n$ checkerboard, several cells are marked such that the bottom-left cell $(L)$ and the top-right cell $(R)$ are not marked, and any knight's path from $L$ to $R$ necessarily passes through a marked cell. For which $n > 3$ can it be definitely stated that there exist three consecutive cells along a diagonal, among which at least two are marked?

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