There are \(90\,000\) five-digit positive integers. However, only some of these five-digit integers satisfy the following conditions:

the middle digit is \(0\),

the ten thousands digit and the tens digit are equal,

the thousands digit and the ones (units) digit are equal, and

the number has exactly \(5\) prime factors. All of these prime factors are odd and none are repeated.

Determine all five-digit positive integers that satisfy the given conditions.