
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 rectangular grid \( M \), there is a dot light, all of which are initially off. In one move, you can draw any line that does not touch any light and turn on all the lights on one side of this line, provided they are all off. Each move must turn on at least one light. The goal is to turn on all the lights using as many moves as possible. Find 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.

Answers: a) 3 moves; b) 4 moves.

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