
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
The set of five-digit numbers \( \{ N_1, \ldots, N_k \} \) is such that any five-digit number with all digits in increasing order coincides in at least one digit with at least one of the numbers \( N_1, \ldots, N_k \). Find the smallest possible value of \( k \).

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