
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
A regular hexagon in the complex plane has vertices \[V=\left\{1, -1, i, -i, \frac{1}{2} + \frac{\sqrt{3}}{2}i, -\frac{1}{2} - \frac{\sqrt{3}}{2}i\right\}.\] For each $j$, $1 \leq j \leq 10$, an element $z_j$ is chosen from $V$ at random, independently of other choices. Let $P = \prod_{j=1}^{10} z_j$ be the product of the 10 numbers selected.

The probability that $P=1+i$ 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}
user