
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
Suppose two arithmetic sequences $\{a\_n\}$ and $\{b\_n\}$ have the sum of their first n terms as $S\_n$ and $T\_n$, respectively. If $a\_3 = 2b\_3$ and $\frac{S\_n}{T\_n} = \frac{3n - t}{2n + 2}$ for any $n \in \mathbb{N}^*$, find the value of $\frac{a\_9}{b\_6 + b\_8} + \frac{a\_5}{b\_3 + b\_11}$.

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