
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
100 bulbs were hung in a row on the New Year's tree. Then the bulbs began to switch according to the following algorithm: all bulbs were lit initially, then after one second, every second bulb was turned off, and after another second, every third bulb switched: if it was on, it was turned off, and if it was off, it was turned on. Again, after one second, every fourth bulb switched, after another second - every fifth bulb, and so on. This procedure continued for 100 seconds. Determine the probability that a randomly chosen bulb is lit after all this (bulbs do not burn out or break).

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