
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 centrally symmetric octagon in the complex plane are defined by 
\[V=\left\{2i, -2i, 1+i, -1+i, 1-i, -1-i, \frac{3}{2}+ \frac{3}{2}i, \frac{3}{2}- \frac{3}{2}i\right\}.\]
For each $j$, $1\leq j\leq 16$, an element $z_j$ is chosen from $V$ at random, independently. Let $P={\prod}_{j=1}^{16}z_j$ be the product of the $16$ numbers selected.

Compute the probability that $P=i$ and represent it 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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