
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
Bob has invented the Very Normal Coin (VNC). When the VNC is flipped, it shows heads  $\textstyle\frac{1}{2}$  of the time and tails  $\textstyle\frac{1}{2}$  of the time - unless it has yielded the same result five times in a row, in which case it is guaranteed to yield the opposite result. For example, if Bob flips five heads in a row, then the next flip is guaranteed to be tails.

Bob flips the VNC an infinite number of times. On the  $n$ th flip, Bob bets  $2^{-n}$  dollars that the VNC will show heads (so if the second flip shows heads, Bob wins  $\$ 0.25 $, and if the third flip shows tails, Bob loses $ \ $0.125$ ).

Assume that dollars are infinitely divisible. Given that the first flip is heads, the expected number of dollars Bob is expected to win can be expressed as  $\frac{a}{b}$  for relatively prime positive integers  $a, b$ . Compute  $100a + b$ .

*Proposed by Lewis Chen*

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