
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
Determine the smallest positive integer  $ n$  such that there exists positive integers  $ a_1,a_2,\cdots,a_n$ , that smaller than or equal to  $ 15$  and are not necessarily distinct, such that the last four digits of the sum,

\[ a_1!\plus{}a_2!\plus{}\cdots\plus{}a_n!\]

Is  $ 2001$ .

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