
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
a) What is the maximum number of rooks that can be placed on a chessboard of $n^{2}$ squares so that no two rooks threaten each other? In how many different ways can this be done?

b) What is the minimum number of rooks that can be placed on a chessboard of $n^{2}$ squares so that these rooks threaten every square on the board? In how many different ways can this be done?

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