
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 square and its diagonals in the complex plane are given by \[ V = \{1, -1, i, -i, \sqrt{2}, -\sqrt{2}, \sqrt{2}i, -\sqrt{2}i\} \]. 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.

Calculate the probability that \( |P| = 2 \).

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