
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
For an increasing sequence of positive integers, if its first term is odd, the second term is even, the third term is odd, the fourth term is even, and so on, then it is called an "alternating sequence." The empty set is also considered an alternating sequence. Let $A(n)$ denote the number of all alternating sequences where each term is taken from the set $\{1, 2, \cdots, n\}$. Find $A(20)$. (1994, British Mathematical Olympiad)

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