
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
There is a set of data: $a_{1}=\frac{3}{1×2×3}$, $a_{2}=\frac{5}{2×3×4}$, $a_{3}=\frac{7}{3×4×5}$, $\ldots $, $a_{n}=\frac{2n+1}{n(n+1)(n+2)}$. Let $S_{n}=a_{1}+a_{2}+a_{3}+\ldots +a_{n}$. Find the value of $S_{12}$. To solve this problem, Xiao Ming first simplified $a_{n}$ to $a_{n}=\frac{x}{(n+1)(n+2)}+\frac{y}{n(n+2)}$, and then calculated the value of $S_{12}$ based on the simplified $a_{n}$. Please follow Xiao Ming's approach to first find the values of $x$ and $y$, and then calculate the value of $S_{12}$.

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