
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
1. Call a year ultra-even if all of its digits are even. Thus \(2000, 2002, 2004, 2006\), and \(2008\) are all ultra-even years. They are all 2 years apart, which is the shortest possible gap. \(2009\) is not an ultra-even year because of the \(9\), and \(2010\) is not an ultra-even year because of the \(1\).

(a) In the years between the years 1 and 10000, what is the longest possible gap between two ultra-even years? Give an example of two ultra-even years that are far apart with no ultra-even years between them. Justify your answer.

(b) What is the second-shortest possible gap (that is, the shortest gap longer than 2 years) between two ultra-even years? Again, give an example, and justify your answer.

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