
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
Given each of the $12$ edges of a rectangular prism is labeled $0$ or $1$, and each of the $6$ rectangular faces must have an even sum of labels from its edges, find the number of such labelings such that the sum of the labels on each of the prism’s faces is exactly $2$.

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