
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  $r_1,r_2,\dots ,r_n$  be real numbers. Given  $n$  reals  $a_1,a_2,\dots ,a_n$  that are not all equal to  $0$ , suppose that inequality
\[r_1(x_1-a_1)+ r_2(x_2-a_2)+\dots + r_n(x_n-a_n)\leq\sqrt{x_1^2+ x_2^2+\dots + x_n^2}-\sqrt{a_1^2+a_2^2+\dots +a_n^2}\]
holds for arbitrary reals  $x_1,x_2,\dots ,x_n$ . Find the values of  $r_1,r_2,\dots ,r_n$ .

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