
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 \(S_{i,j} \subseteq \{1, 2, \ldots, 2n\}\) (for \(i, j \in \{0, 1, \ldots, n\}\)) that satisfy the following two conditions:
1. For any \(i, j \in \{0, 1, \ldots, n\}\), the set \(S_{i,j}\) contains exactly \(i+j\) elements.
2. For any \(0 \leq i \leq k \leq n\) and \(0 \leq j \leq t \leq n\), we have \(S_{i,j} \subseteq S_{k,t}\).
Determine the number of different ways to make such selections.

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