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"found out through" is correct and usable in written English.
You can use it when you are referring to a method, source, or means of discovering something. For example: "We found out through our research that the building was constructed in 1900."
Exact(60)
"We found out through a newspaper," he added, referring to an article published in The Villager.
"We'd already found out through all our other sources that there were at best two confirmed entrants".
Military secrets can be concealed by hush-hush governmental tactics and found out through espionage, but secrets of nature are discoverable in laboratories anywhere in the world.
In 1999, before Woods hired Williams, Navarro found out through back channels that the job was his if he wanted it.
We found out, through experiments, that the GA based approach detects LS earlier than the Markov based approach, and consequently provides personalized content earlier, making the learning process easier.
Many likely found out through him.
"We just found out, through ancestry.com, that we're both part-Jewish," she said.
(Read this and this for more background) The pair only found out through their own efforts.
Instead she and her friends found out through their own detective work and a chance discovery.
I found out through the press that Mr Nelsinho was going to get immunity.
Once she found out through her own research, she was angry.
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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