
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
Consider a centrally symmetric octagon in the complex plane with vertices \(V = \left\{ \pm\sqrt{2}i, \pm 1, \frac{1}{\sqrt{8}}(1+i), \frac{1}{\sqrt{8}}(-1+i), \frac{1}{\sqrt{8}}(1-i), \frac{1}{\sqrt{8}}(-1-i) \right\}\). For each \(j\), \(1 \leq j \leq 16\), an element \(z_j\) is chosen from \(V\) at random, independently of the other choices. Let \(P = \prod_{j=1}^{16} z_j\) be the product of the 16 numbers selected. Determine the probability that \(P = -1\) and express it in the form \(\frac{a}{p^b}\), where \(a\), \(b\), and \(p\) are positive integers with \(p\) prime and \(a\) not divisible by \(p\). Find \(a + b + p\).

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