
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
Suppose
$$
\frac{1}{2013^{1000}}=a_{1}+\frac{a_{2}}{2!}+\frac{a_{3}}{3!}+\cdots+\frac{a_{n}}{n!},
$$
where \( n \) is a positive integer, \( a_{1}, \ldots, a_{n} \) are nonnegative integers such that \( a_{k} < k \) for \( k = 2, \ldots, n \) and \( a_{n} > 0 \). Find the value of \( n \).

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