
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 that includes the consecutive digit string "11235" in its decimal representation. Let \( k \) be a positive integer such that \( 10^k > N \). For all such valid positive integers \( N \) and \( k \), find the minimum possible value of \( \frac{10^k - 1}{\operatorname{gcd}(10^k - 1, N)} \).

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