
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 \(\left\{a_n\right\}\) that satisfies \(a_1 = a_2 = 1\) and the recurrence relation \(a_{n+1} a_{n-1} = a_n^2 + n a_n a_{n-1} \) for \(n=2,3, \ldots\):
1. Find the general formula for the sequence \(\left\{a_n\right\}\).
2. Compute \(\sum_{k=2}^{n} \frac{a_k}{(k-2)!}\).

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