
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
Let \( N \) be a positive integer whose decimal representation contains 11235 as a contiguous substring, and let \( k \) be a positive integer such that \( 10^k > N \). Find the minimum possible value of
\[
\frac{10^k - 1}{\operatorname{gcd}(N, 10^k - 1)}
\]


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