
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 $s$ be one of the real solutions to the equation $x^4 - x - \frac{1}{2} = 0$. Find the exact numerical value of:
\[
T = s^3 + 2s^7 + 3s^{11} + 4s^{15} + \dotsb.
\]

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