
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  $x_1, x_2 \ldots , x_5$  be real numbers. Find the least positive integer  $n$  with the following property: if some  $n$  distinct sums of the form  $x_p+x_q+x_r$  (with  $1\le p<q<r\le 5$ ) are equal to  $0$ , then  $x_1=x_2=\cdots=x_5=0$ .

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