
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

In the center of each cell of a grid rectangle, there is a small light bulb, all initially turned off. In each move, it is allowed to draw any straight line that does not touch the bulbs, and turn on all the bulbs on one side of this line if all of them are off. Each move must turn on at least one bulb. The task is to turn on all the bulbs, making as many moves as possible. What is the maximum number of moves that can be made if:

a) the grid is a $21 \times 21$ square

b) the grid is a $20 \times 21$ rectangle

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