
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
Max has a light bulb and a defective switch. The light bulb is initially off, and on the  $n$ th time the switch is flipped, the light bulb has a  $\tfrac 1{2(n+1)^2}$  chance of changing its state (i.e. on  $\to$  off or off  $\to$  on). If Max flips the switch 100 times, find the probability the light is on at the end.

*Proposed by Connor Gordon*

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