
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
Determine the maximal size of a set of positive integers with the following properties: $1.$  The integers consist of digits from the set  $\{ 1,2,3,4,5,6\}$ . $2.$  No digit occurs more than once in the same integer. $3.$  The digits in each integer are in increasing order. $4.$  Any two integers have at least one digit in common (possibly at different positions). $5.$  There is no digit which appears in all the integers.

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