
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
There is a cube with edge length \( a \), where one of its faces is on \( P_1 \) and another face is on \( P_2 \). The vertices of the cube are numbered \( 1, 2, \ldots, 8 \). Transform the cube according to \( x_{14} \) and \( x_{45} \) and determine the mathematical relationships between \( a \), \( \overline{12}^{V} \overline{14}^{V} \), and \( \overline{15}^{V} \).

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