
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
6. A triple of integers  $(a,b,c)$  is said to be  $\gamma$ *-special* if  $a\le \gamma(b+c)$ ,  $b\le \gamma(c+a)$  and  $c\le\gamma(a+b)$ . For each integer triple  $(a,b,c)$  such that  $1\le a,b,c \le 20$ , Kodvick writes down the smallest value of  $\gamma$  such that  $(a,b,c)$  is  $\gamma$ -special. How many distinct values does he write down?

*Proposed by winnertakeover*

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