
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
An increasing sequence of integers is called a zigzag sequence if its first term is odd, its second term is even, its third term is odd, its fourth term is even, and so on. The empty set is also considered a zigzag sequence. Let $A(n)$ denote the number of all zigzag sequences where each term is taken from the set $\{1, 2, \cdots, n\}$. It is evident that $A(1)=2$ and $A(2)=3$. Find $A(20)$ and provide a justification for your answer.

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