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"for the first run" is correct and usable in written English.
It is an idiomatic expression and can be used when referring to an initial attempt or trial of something. For example, "We ran the experiment for the first run and got promising results."
Exact(58)
Ryan Flaherty takes Kuroda deep for the first run of the game.
J.D. Drew, Fernando Viña and Edgar Renteria hit consecutive singles for the first run.
Others might have been unsettled with the 50th starting position for the first run.
Rey Sanchez and Mike Sweeney doubled to lead off the inning for the first run.
In America, 400,000 copies were ordered for the first run, and another 350,000 for the second.
The list of those who qualify for the first run is expected to include some familiar names.
The error provided an opportunity for the Angels, and Troy Glaus smashed a single to left for the first run.
Rockies starter Greg Smith hit a double into the left-field corner to score Barmes for the first run batted in of his career.
"The Sopranos" reached as many as 13.4 million viewers for the first run of an individual episode, and several other times topped 12 million.
Easley drove Armando Almanza's first pitch into the left-field seats for the first run against Florida's bullpen in 15 innings.
Sensory's engineers want to buy circuit boards from a Taiwanese company and mount their voice-recognition chip on them, at least for the first run of robotic dogs.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com