
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
The three roots of the cubic  $ 30 x^3 \minus{} 50x^2 \plus{} 22x \minus{} 1$  are distinct real numbers between  $ 0$  and  $ 1$ .  For every nonnegative integer  $ n$ , let  $ s_n$  be the sum of the  $ n$ th powers of these three roots.  What is the value of the infinite series
\[ s_0 \plus{} s_1 \plus{} s_2 \plus{} s_3 \plus{} \dots \, ?\]

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