Three prime numbers, p₁, p₂, p₃, are processed by multiplying the first by 7, the second by 8 and the third by 9, so you have

7p₁, 8p₂ and 9p₃.

and it's found that

7p₁ < 8p₂ < 9p₃.

Taken together, 7p₁, 8p₂ and 9p₃ use all nine digits 1 - 9 exactly once each, with no zeros.

Also, the product of 7p₁, 8p₂ and 9p₃ is a 9-digit number containing each digit 1 - 9.

What are the three processed primes, 7p₁, 8p₂ and 9p₃?