
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 group of vertices \(V\) of a centrally symmetric octagon in the complex plane, represented by \[V=\{\sqrt{3}i,-\sqrt{3}i, \frac{3}{\sqrt{18}}(1+i), \frac{3}{\sqrt{18}}(-1+i), \frac{3}{\sqrt{18}}(1-i), \frac{3}{\sqrt{18}}(-1-i), \frac{1}{\sqrt{18}}(1+i), \frac{1}{\sqrt{18}}(-1-i)\}\]. For each \(j\), \(1\leq j\leq 16\), an element \(z_j\) is randomly chosen from \(V\), independently of the other choices. Let \(Q = \prod_{j=1}^{16} z_j\) be the product of the \(16\) numbers chosen.

The probability that \(Q=i\) can be expressed in the form
\[\frac{a}{p^b},\] where \(a,\) \(b,\) \(p\) are integers, \(p\) a prime, and \(a\) not divisible by \(p.\) Find \(a + b + p.\)

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