英语翻译Ponder This Challenge:N lamps are set in a circle,and fo

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英语翻译
Ponder This Challenge:
N lamps are set in a circle,and for each integer M you have a tool that can toggle the state (on/off) of any set of M consecutive lamps.
Find a possible N which satisfies the following statements:
The sum of its digits is less than 10.
By applying the tool for M=105 several times,we can toggle a single lamp.
If we remove one lamp and start from a random initial setting for the remaining N-1 lamps,the probability that there exists a way to apply the tool for M=32 several times and switch all the lamps off is less than 0.001%.

思考这个挑战：
n灯设置在一个圆圈,并为每个整数m,你有一个工具,可以切换状态（开/关）的任何一组的M个连续灯.
找到一个可能的n满足下列语句：
的数字之和小于10.
通过应用工具,M = 105几次,我们可以切换单灯.
如果我们从剩余N-1灯一个随机的初始设置中删除一个灯开始,可能存在一种方法适用于m = 32多次的工具和开关所有的灯关闭小于0.001%.

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