
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
If the sets $A_1, A_2, \ldots, A_n$ satisfy $A_1 \cup A_2 \cup \ldots \cup A_n = A$, then $A_1, A_2, \ldots, A_n$ are considered a partition of set $A$. It is known that:
1. When $A_1 \cup A_2 = \{a_1, a_2, a_3\}$, there are $3^3$ partitions;
2. When $A_1 \cup A_2 \cup A_3 = \{a_1, a_2, a_3, a_4\}$, there are $7^4$ partitions;
3. When $A_1 \cup A_2 \cup A_3 \cup A_4 = \{a_1, a_2, a_3, a_4, a_5\}$, there are $15^5$ partitions;
...
From the above conclusions, infer the general conclusion:
When $A_1 \cup A_2 \cup \ldots A_n = \{a_1, a_2, a_3, \ldots a_{n+1}\}$, there are     partitions.

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