
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
Find the smallest positive integer  $n$ , such that there exist  $n$  integers  $x_1, x_2, \dots , x_n$  (not necessarily different), with  $1\le x_k\le n$ ,  $1\le k\le n$ , and such that
\[x_1 + x_2 + \cdots + x_n =\frac{n(n + 1)}{2},\quad\text{ and }x_1x_2 \cdots x_n = n!,\]
but  $\{x_1, x_2, \dots , x_n\} \ne \{1, 2, \dots , n\}$ .

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