
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
A sequence of $A$'s and $B$'s is called antipalindromic if writing it backwards, then turning all the $A$'s into $B$'s and vice versa, produces the original sequence. For example, $ABBAAB$ is antipalindromic. For any sequence of $A$'s and $B$'s, we define the cost of the sequence to be the product of the positions of the $A$'s. For example, the string $ABBAAB$ has cost $1 \cdot 4 \cdot 5 = 20$. Find the sum of the costs of all antipalindromic sequences of length 2020.

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