
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 a non-negative integer \( n \). Select \( (n+1)^{2} \) sets \( S_{i, j} \subseteq \{1, 2, \dots, 2n\} \) for \( i, j = 0, 1, \dots, n \) satisfying the following two conditions:
1. For any \( i, j \in \{0, 1, \dots, n\} \), the set \( S_{i, j} \) has \( i+j \) elements.
2. For any \( 0 \leq i \leq k \leq n \) and \( 0 \leq j \leq l \leq n \), \( S_{i, j} \subseteq S_{k, l} \).

Determine the number of different ways to select these sets.

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