
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 array \( \mathrm{I} \):

\[
\begin{array}{cccccccccc} 
& 1 & 2 & 3 & 4 & \cdots & n-3 & n-2 & n-1 & \\
1 & 2 & 3 & 4 & 5 & \cdots & n-2 & n-1 & n \\
2 & 3 & 4 & 5 & 6 & \cdots & n-1 & n & & (n \geqslant 2),
\end{array}
\]

Select a number from each column \( k (k=1,2, \cdots, n) \), denoted as \( a_{k} \). If \( a_{1}, a_{2}, \cdots, a_{n} \) are all distinct, then the ordered \( n \)-tuple \(\left(a_{1}, a_{2}, \cdots, a_{n}\right)\) is called a system of distinct representatives (SDR) of array \( \mathrm{I} \), where \( a_{k} \) is referred to as the representative of column \( k \). Let \( u_{n} \) denote the number of SDRs of array \( \mathrm{I} \). Find \( u_{n} \).

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