Let a>1 be a positive integer. The sequence of natural numbers \left\{a_{n}\right\} is defined as follows: a_{1}=a and for all n \geq 1, a_{n+1} is the largest prime factor of a_{n}^{2}-1. Determine the smallest possible value of a such that the numbers a_{1}, a_{2}, \ldots, a_{7} are all distinct.