
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: 
\[W=\left\{\sqrt{2}i, -\sqrt{2}i, \sqrt{2}, -\sqrt{2}, \frac{1}{\sqrt{8}}(1+i), \frac{1}{\sqrt{8}}(-1+i), \frac{1}{\sqrt{8}}(1-i), \frac{1}{\sqrt{8}}(-1-i) \right\}.\]
For each $k$, $1\leq k \leq 16$, an element $w_k$ is chosen from $W$ at random, independently of the other choices. Let $Q=\prod_{k=1}^{16}w_k$ be the product of the $16$ numbers selected.

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

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