Andrew generates a finite random sequence \{a_{n}\} of distinct integers according to the following criteria:
- a_{0}=1,0<\left|a_{n}\right|<7 for all n, and a_{i} \neq a_{j} for all i\lt{j}.
- a_{n+1} is selected uniformly at random from the set \{a_{n}-1, a_{n}+1,-a_{n}\}, conditioned on the above rule. The sequence terminates if no element of the set satisfies the first condition.
For example, if \left(a_{0}, a_{1}\right)=(1,2), then a_{2} would be chosen from the set \{-2,3\}, each with probability \frac{1}{2}. Determine the probability that there exists an integer k such that a_{k}=6.