Unlocking the Gemini Secret

Under the night sky, the Gemini stars twinkle, said to hide a secret of the universe. Astronomer Xiao Lan examines an old star chart through his antique telescope and finds two annotated numbers: \(20255202\) and \(10244201\). Next to them, a note reads: if there exists a positive integer \(n\) such that both \(n+20255202\) and \(n+10244201\) are perfect squares, then the message of the Gemini can be unlocked.

A number is a perfect square if it can be written as \(m^2\) for some integer \(m\) (for example, \(1=1^2\), \(4=2^2\), \(9=3^2\), etc.).

Your task is to help Xiao Lan determine the number of positive integers \(n\) that satisfy the condition.

Hint: Let \(n+20255202=x^2\) and \(n+10244201=y^2\). Then \(x^2-y^2=20255202-10244201=10111001\). Factor this difference as \((x-y)(x+y)=10111001\) and find all divisor pairs \((d,e)\) with the same parity (and \(d0\>.

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This problem does not require any input from the user.

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Output a single integer representing the count of positive integers \(n\) that satisfy the condition.

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