
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
In the sequence $\{a_n\}$, where $a_1=\frac{1}{2}$ and $a_{n+1}=\frac{a_n}{1+3a_n}$, determine the value of $T_8$, where $T_n$ denotes the sum of the first $n$ terms of the sequence $\{\frac{1}{a_n}\}$.

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