
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's call a natural number "good" if all its digits are nonzero. A good number is called "special" if it has at least \( k \) digits and the digits are in strictly increasing order (from left to right).

Suppose we have a good number. In one move, it is permissible to either append or insert a special number at any edge or between any two digits of the good number, or conversely, to remove a special number from the good number. For what maximum \( k \) can any good number be transformed into any other good number using such moves?

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