
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 nonnegative integer. Determine the number of ways to choose sets \( S_{ij} \subseteq \{1, 2, \ldots, 2n\} \), for all \( 0 \leq i \leq n \) and \( 0 \leq j \leq n \) (not necessarily distinct), such that:
- \(\left| S_{ij} \right| = i + j \), and
- \( S_{ij} \subseteq S_{kl} \) if \( 0 \leq i \leq k \leq n \) and \( 0 \leq j \leq l \leq n \).

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