
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  $r$  be a fixed positive real number. It is known that for some positive integer  $n$  the following statement is true: for any positive real numbers  $a_1,...,a_n$  satisfying the equation  $a_1+...+a_n=r(\frac{1}{a_1}+...+\frac{1}{a_n})$  they also satisfy the equation  $\frac{1}{\sqrt{r}-a_1}+...+\frac{1}{\sqrt{r}-a_n}=\frac{1}{\sqrt{r}}$  ( $a_i\neq\sqrt{r}$ ). Find  $n$ .

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