
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 a  $9$ -digit number be balanced if it has all numerals  $1$  to  $9$ . Let  $S$  be the sequence of the numerals which is constructed by writing all balanced numbers in increasing order consecutively. Find the least possible value of  $k$  such that any two subsequences of  $S$  which has consecutive  $k$  numerals are different from each other.

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