
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 two arithmetic sequences $\{a\_n\}$ and $\{b\_n\}$ with the sum of their first $n$ terms being $S\_n$ and $T\_n$, respectively. If $\frac{S\_n}{T\_n} = \frac{2n-3}{4n-1}$ holds for any natural number $n$, find the value of $\frac{a\_3+a\_{15}}{2(b\_3+b\_9)} + \frac{a\_3}{b\_2+b\_{10}}$.

A) $\frac{19}{43}$
B) $\frac{17}{40}$
C) $\frac{9}{20}$
D) $\frac{27}{50}$

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