
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
Six real numbers  $x_1<x_2<x_3<x_4<x_5<x_6$  are given. For each triplet of distinct numbers of those six Vitya calculated their sum. It turned out that the  $20$  sums are pairwise distinct; denote those sums by   $$ s_1<s_2<s_3<\cdots<s_{19}<s_{20}. $$   It is known that  $x_2+x_3+x_4=s_{11}$ ,  $x_2+x_3+x_6=s_{15}$  and  $x_1+x_2+x_6=s_{m}$ . Find all possible values of  $m$ .

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