
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 $W = \left\{2i, -2i, 1+i, -1-i, 1-i, -1+i, \sqrt{2}+i\sqrt{2}, -\sqrt{2}-i\sqrt{2}\right\}$. An element $u_j$ for each $j$, $1 \leq j \leq 16$, is chosen randomly from $W$ with independent selections. Let $Q = \prod_{j=1}^{16} u_j$ be the product of the 16 numbers chosen.

Calculate the probability that $Q = 1$, expressing your result in the form $\frac{c}{q^d}$ where $c$, $d$, and $q$ 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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