
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
Given an integer  $n\geq 2$ . There are  $N$  distinct circle on the plane such that any two circles have two distinct intersections and no three circles have a common intersection. Initially there is a coin on each of the intersection points of the circles. Starting from  $X$ , players  $X$  and  $Y$  alternatively take away a coin, with the restriction that one cannot take away a coin lying on the same circle as the last coin just taken away by the opponent in the previous step. The one who cannot do so will lost. In particular, one loses where there is no coin left. For what values of  $n$  does  $Y$  have a winning strategy?

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