
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 the positive real solution to $x^4 + \frac{1}{3}x - 1 = 0$. Find the exact numerical value of:
\[ S = r^3 + 2r^7 + 3r^{11} + 4r^{15} + \dotsb. \]

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