
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
In how many ways can the power set of an $n$-element set be partitioned into two disjoint sets, denoted as $I$ and $H$, such that the following conditions are simultaneously satisfied?

a) For any $a, b \in I$, both $a \cup b \in I$ and $a \cap b \in I$.

b) For any $a, b \in H$, both $a \cup b \in H$ and $a \cap b \in H$.

c) For any $a \in I$ and $b \in H$, both $a \cup b \in I$ and $a \cap b \in H$.

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