125 and 512; 256 and 625

For how many positive integers \(n<1000\), do there exist integers \(a,\) \(b,\) \(c,\) and \(d,\) that satisfy the following conditions:

\[\begin{cases} 100a+b =c^n\\ 10b+a=d^n\\ gcd(c,d)=1? \end{cases}\]

Details and assumptions

There are no restrictions on the values of \(a\) and \(b\).

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