
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

There are \( n+1 \) points on a line \( l \), numbered from left to right as \( 1, 2, \cdots, n+1 \). In the plane above \( l \), these points are connected by \( n \) continuous curves meeting the following conditions:
1. Each curve connects exactly two different points, and there is at most one curve between any two points.
2. If point \( i \) is connected to point \( j \), denote the connection as \( l_{ij} \) where \( i < j \).
3. For \( i < m \leqslant j < n \), \( l_{ij} \) and \( l_{mn} \) cannot both exist, which means they cannot "cross" or "connect end-to-end".

Let the number of different ways to connect these points be denoted as \( A_n \) for \( n \geqslant 0, n \in \mathbf{Z} \). Find the expression for \( A_n \).

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