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

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