
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 fixed positive integer. For real \( \alpha \), define the sequence \( x_k \) by: \( x_0 = 0 \), \( x_1 = 1 \), \( x_{k+2} = \frac{\alpha x_{k+1} - (N - k)x_k}{k + 1} \). Find the largest \( \alpha \) such that \( x_{N+1} = 0 \) and the resulting \( x_k \).

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