
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 given by \[ V=\left\{ \sqrt{3}i, -\sqrt{3}i, \frac{1}{\sqrt{9}}(1+i), \frac{1}{\sqrt{9}}(-1+i), \frac{1}{\sqrt{9}}(1-i), \frac{1}{\sqrt{9}}(-1-i), \frac{1}{\sqrt{9}}(2+i), \frac{1}{\sqrt{9}}(-2-i) \right\} \]. For each \( j \), \( 1 \leq j \leq 14 \), an element \( z_j \) is chosen from \( V \) at random, independently of the other choices. Let \( P={\prod}_{j=1}^{14}z_j \) be the product of the 14 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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