I open a notepad after the first two lines if it isn't obvious what it is by then.
And I lay out the possibilities
R O _ _ _
_ R O _ _
R _ O _ _
etc
and I see what words I can build with the letters I haven't used yet. Rowdy jumped right out at me from the first line, and that meant that neither R nor O could be in the first position.
so the grid became
_ R O _ Y
_ _ O R Y
_ R _ O Y
Because R could only be in the second and fifth position, and O could only be in the 4th and 5th. With S, I, E, C, T, A, L, W, and D out of the picture, I spent a lot of time trying to figure out if U fit somewhere in there, but nothing I came up with was possible without one of the greyed out letters. The X threw me off because, as its scrabble score indicates, it is very uncommonly used.
But after screwing around in notepad for 5 minutes at that point, it became clear the best shot was PROXY. And it was.