
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 the equation $x^3 - \frac{1}{4}x - 1 = 0$. Find the exact numerical value of the infinite series:
\[ T = r + 2r^4 + 3r^7 + 4r^{10} + \dotsb. \]

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