
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
Let $n$ be a positive integer. Define the sequence $(y_k)$ by $y_0 = 0,$ $y_1 = 1,$ and
\[y_{k + 2} = \frac{(n - 1) y_{k + 1} + (n + k) y_k}{k + 1}\] for $k \ge 0.$  Find the sum $y_0 + y_1 + y_2 + \dotsb + y_{2n}$ as a function of $n.$

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