
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 set of vertices \( V \) in the complex plane, given by 
\[ V = \left\{ \sqrt{3}i, -\sqrt{3}i, \frac{\sqrt{2}}{2}(1+i), \frac{\sqrt{2}}{2}(-1+i), \frac{\sqrt{2}}{2}(1-i), \frac{\sqrt{2}}{2}(-1-i) \right\}. \]
For each \( j \), \( 1 \leq j \leq 10 \), an element \( z_j \) is chosen from \( V \) at random, independently of the other choices. Let \( P = \prod_{j=1}^{10} z_j \) be the product of the 10 numbers selected. Find the probability that \( P = 1 \) and the product of the chosen elements with magnitude \( \sqrt{3} \) is \( -1 \).

Present the answer in LaTex format: \boxed{Your answer}
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