
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
Some bishops and knights are placed on an infinite chessboard, where each square has side length  $1$  unit. Suppose that the following conditions hold:


- For each bishop, there exists a knight on the same diagonal as that bishop (there may be another piece between the bishop and the knight).
- For each knight, there exists a bishop that is exactly  $\sqrt{5}$  units away from it.
- If any piece is removed from the board, then at least one of the above conditions is no longer satisfied.

If  $n$  is the total number of pieces on the board, find all possible values of  $n$ .

*Sheldon Kieren Tan*

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