
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 largest positive integer  $n$  such that the following statement holds:
If  $a_1,a_2,a_3,a_4,a_5,a_6$  are six distinct positive integers less than or equal to  $n$ , then there exist  $3$  distinct positive integers ,from these six, say  $a,b,c$  s.t.  $ab>c,bc>a,ca>b$ .

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