Random Walk on a Telephone Pad: Divisibility by 2

Consider a standard telephone number pad arranged as: ``` 1 2 3 4 5 6 7 8 9 ``` You press a random digit (uniformly from 1-9), write it down, then move to a uniformly random orthogonally adjacent digit (up, down, left, or right -- no diagonals), write that down, then move again to a random adjacent digit and write it down. The three digits form a 3-digit number (e.g., if you press 5, then 6, then 9, the number is 569). What is the probability that this 3-digit number is divisible by 2?