
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 a sequence $\{a_n\}$ where each term is a positive number, the first term $a_1=1$, and $S_n$ is the sum of the first $n$ terms of the sequence $\{a_n\}$. The sequence satisfies the condition: $a_nS_{n+1}-a_{n+1}S_n+a_n-a_{n+1}=\frac{1}{2}a_na_{n+1}$. Find the value of $\frac{3}{34}S_{12}$.

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