
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
From an array of \( n^2 \) light bulbs, a \( n \times n \) display is assembled. Each light bulb has two states - on and off. When pressing any given light bulb, its state is kept the same, but all the light bulbs in the same row or column change their state to the opposite. Initially, all the light bulbs on the display are turned off. Vasya sequentially pressed several light bulbs, with no two pressed light bulbs being in the same row or column. What is the maximum number of light bulbs Vasya could have turned on?

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