
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
For a positive integer \( k \), find the greatest common divisor (GCD) \( d \) of all positive even numbers \( x \) that satisfy the following conditions:
1. Both \( \frac{x+2}{k} \) and \( \frac{x}{k} \) are integers, and the difference in the number of digits of these two numbers is equal to their difference;
2. The product of the digits of \( \frac{x}{k} \) is a perfect cube.

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