
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 $P_1, P_2, \ldots, P_j$ be subsets of the set $P = \{1, 2, \ldots, i\}$, where $i, j$ are positive integers. Let $a_{ij}$ denote the number of ordered subset groups $(P_1, P_2, \ldots, P_j)$ that satisfy $P_1 \cap P_2 \cap \ldots \cap P_j = \emptyset$.
1. Find the value of $a_{22}$;
2. Find the expression for $a_{ij}$.

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