
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
Let \( n \) be a non-negative integer. Determine the number of ways to select a family of sets \( S_{i,j} \subseteq \{1, 2, \ldots, 2n\} \) (where \( 0 \leq i \leq n \) and \( 0 \leq j \leq n \), and repetitions are allowed) such that:
1. \( \left|S_{i,j}\right| = i + j \);
2. If \( 0 \leq i \leq k \leq n \) and \( 0 \leq j \leq l \leq n \), then \( S_{i,j} \subseteq S_{k,l} \).

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