
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 $n \times n$ chessboard, where $n \geq 3$ is an integer, contains a lamp. Initially, the lamps in two opposite corners are turned on, and the others are off. An operation consists of selecting a row or a column of the chessboard and changing the state of all the lamps in that row or column.

Before starting the operations, Alice can choose to individually turn on as many lamps as she wants. She wants to ensure that there exists a finite sequence of operations after which each lamp is off. How many lamps, at minimum, must she turn on before the start of the operations?

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