
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
Let  $k \geq 1$  and  $N>1$  be two integers. On a circle are placed  $2N+1$  coins all showing heads. Calvin and Hobbes play the following game. Calvin starts and on his move can turn any coin from heads to tails. Hobbes on his move can turn at most one coin that is next to the coin that Calvin turned just now from tails to heads. Calvin wins if at any moment there are  $k$  coins showing tails after Hobbes has made his move. Determine all values of  $k$  for which Calvin wins the game. 

*Proposed by Tejaswi Navilarekallu*

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