
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
In an $n \times n$ grid, some cells are colored black, but the cell at the bottom-left corner $L$ and the cell at the top-right corner $R$ are not colored black. On every possible path for a knight from $L$ to $R$, there is at least one black cell. For what values of $n > 3$ can we assert that there exists a set of three diagonally connected cells, at least two of which are colored black?

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