
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 the  $n\times n$  checker board, several cells were marked in such a way that lower left ( $L$ ) and upper right( $R$ ) cells are not marked and that for any knight-tour from  $L$  to  $R$ , there is at least one marked cell. For which  $n>3$ , is it possible that there always exists three consective cells going through diagonal for which at least two of them are marked?

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