
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 a string of \(P\) 's and \(Q\) 's, the value is defined to be the product of the positions of the \(P\) 's. For example, the string \(P P Q P Q Q\) has value \(1 \cdot 2 \cdot 4 = 8\).

Also, a string is called antipalindromic if writing it backwards, then turning all the \(P\) 's into \(Q\) 's and vice versa, produces the original string. For example, \(P P Q P Q Q\) is antipalindromic.

There are \(2^{1002}\) antipalindromic strings of length 2004. Find the sum of the reciprocals of their values.

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