
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
Since counting the numbers from 1 to 100 wasn't enough to stymie Gauss, his teacher devised another clever problem that he was sure would stump Gauss. Defining  $\zeta_{15} = e^{2\pi i/15}$  where  $i = \sqrt{-1}$ , the teacher wrote the 15 complex numbers  $\zeta_{15}^k$  for integer  $0 \le k < 15$  on the board. Then, he told Gauss:

On every turn, erase two random numbers  $a, b$ , chosen uniformly randomly, from the board and then write the term  $2ab - a - b + 1$  on the board instead. Repeat this until you have one number left. What is the expected value of the last number remaining on the board?

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