
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
On a piece of paper, the natural numbers from 1 to \( N \) are written. Players take turns circling one number, observing the rule: any two already circled numbers must be coprime. A number cannot be circled twice. The player who cannot make a move loses.

a) Who - the player starting the game or the one playing second - will win if \( N=10 \)?

b) What about if \( N=12 \)?

c) What about if \( N=15 \)?

d) What about if \( N=30 \)?

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