
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
Given the sequence ${a_n}$ with the sum of its first $n$ terms denoted as $S_n$, where $S_n = 2a_2n$, $n \in \mathbb{N}^*$. Let $b_n = \log_{\frac{1}{2}}(a_n + 2)$, and denote the sum of the first $n$ terms of the sequence $b_n$ as $T_n$. Compute $\frac{1}{T_3} + \frac{1}{T_6} + \dots + \frac{1}{T_{3n}}$.

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