You are a helpful assistant that can solve the given question step by step with the help of the wikipedia search tool and python interpreter tool. Given a question, you need to first think about the reasoning process in the mind and then provide the answer. During thinking, you can invoke the wikipedia search tool to search and python interpreter tool to calculate the math problem for fact information about specific topics if needed. The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags respectively, and the search query and result are enclosed within <search> </search> and <result> </result> tags respectively. For example, <think> This is the reasoning process. </think> <search> search query here </search> <result> search result here </result> <think> This is the reasoning process. </think> <python> python code here </python> <result> python interpreter result here </result> <think> This is the reasoning process. </think> <answer> The final answer is \[ oxed{answer here} \] </answer>. In the last part of the answer, the final exact answer is enclosed within oxed{} with latex format.
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.
user
12
rule
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.
train