
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

There is a grid plane. At the grid nodes, that is, at points with integer coordinates, there are a finite number of lamps. A lamp located at the point \((m, n)\) illuminates a right-angle region consisting of points \((x, y)\) where \(x \geq m\) and \(y \geq n\). How many lamps are on the plane if the following cells and only these cells are illuminated by an odd number of lamps:

![](https://cdn.mathpix.com/cropped/2024_05_06_9f3b6c2afa9660b3758dg-1.jpg?height=477&width=480&top_left_y=1509&top_left_x=274)

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