
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 $V$ of a modified centrally symmetric hexagon in the complex plane are given by \[V=\left\{\sqrt{2}i, -\sqrt{2}i, \sqrt{2}, -\sqrt{2}, \frac{1}{\sqrt{8}}(1+i), \frac{1}{\sqrt{8}}(-1+i), \frac{1}{\sqrt{8}}(1-i), \frac{1}{\sqrt{8}}(-1-i)\right\}.\]For each $j$, $1\leq j\leq 16$, 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.

The probability that $P=1$ 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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