
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
The vertices $W$ of a centrally symmetric octagon in the complex plane are given by \[W=\left\{2i, -2i, \frac{1}{2}(1+i), \frac{1}{2}(-1+i), \frac{1}{2}(1-i), \frac{1}{2}(-1-i), 1+i, 1-i\right\}.\] For each $j$, $1\leq j\leq 16$, an element $w_j$ is chosen from $W$ at random, independently of the other choices. Let $Q=\prod_{j=1}^{16} w_j$ be the product of the $16$ numbers selected.

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

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