Prime digit replacements

By replacing the \( 1^{st} \) digit of the \( 2 \)-digit number \( *3 \), it turns out that six of the nine possible values: \( 13 \), \( 23 \), \( 43 \), \( 53 \), \( 73 \), and \( 83 \), are all prime.

By replacing the \( 3 \)rd and \( 4 \)th digits of \( 56 \)*\( * \)3 with the same digit, this \( 5 \)-digit number is the first example having seven primes among the ten generated numbers, yielding the family: \( 56003 \), \( 56113 \), \( 56333 \), \( 56443 \), \( 56663 \), \( 56773 \), and \( 56993 \). Consequently \( 56003 \), being the first member of this family, is the smallest prime with this property.

Find the smallest prime which, by replacing part of the number (not necessarily adjacent digits) with the same digit, is part of an eight prime value family.