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

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