
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
Each cell of an $8 \times 8$ board is occupied by a guard. Each guard can look in one of four possible directions (along the lines of the board) and can watch all the guards in the line of their sight. For what maximum $k$ can we direct the guards' gaze such that every guard is watched by at least $k$ other guards?

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