
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  $\alpha < 0 < \beta$  and consider the polynomial  $f(x) = x(x-\alpha)(x-\beta)$ . Let  $S$  be the set of real numbers  $s$  such that  $f(x) - s$  has three different real roots. For  $s\in S$ , let  $p(x)$  the product of the smallest and largest root of  $f(x)-s$ . Determine the smallest possible value that  $p(s)$  for   $s\in S$ .

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