
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
At the center of each cell of a checkered rectangle \( M \), there is a tiny light bulb, which are all initially turned off. During each move, you are allowed to draw any line that does not touch any light bulbs and turn on all light bulbs on one side of this line if they are all off. Each move must turn on at least one light bulb. You need to turn on all the light bulbs by making as many moves as possible. 

What is the maximum number of moves you can make if

a) \(M\) is a \(21 \times 21\) square;

b) \(M\) is a \(20 \times 21\) rectangle?

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