I have been practicing different routing exercises and came upon a Dijkstra question which I am unable to decide how to proceed.

My question is which node do i pick first? Since the path is the same: $u-w= 6 $ and $u-x =6.$
Any solutions are appreciated.
Edit: Solution provided by a colleague of mine. Is this the way to go?
The first chosen edge doesn't matter -- no matter which one is picked, the other one is picked immediately afterwards. However, if order is really really important to you, you could just assert alphabetical ordering -- tiebreaks happen by alphabetical order.