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The phrase "accepted the fact" is correct and usable in written English.
It can be used when someone acknowledges or comes to terms with a particular reality or situation.
Example: "After much contemplation, she finally accepted the fact that she would need to change her career path."
Alternatives: "acknowledged the reality" or "came to terms with the truth."
Exact(57)
There we have accepted the fact that risks are inevitable".
He just accepted the fact that Pill had coördinated his doom.
His teammates long ago accepted the fact that Knoblauch is going to make throwing errors.
Aaron White accepted the fact that those friends of his had indeed turned on him.
"I've accepted the fact that there are various personages in there," he said about himself.
Her parents had just been divorced and he accepted the fact blandly.
"They haven't accepted the fact," Mr. Diaz de la Portilla said, "they lost the 2000 election".
"They don't like it, but they've accepted the fact that they're just tense people".
"I had accepted the fact that I wasn't going to play him," Mr. Langella said.
Similar(2)
"We established the facts and … my superiors at the company accepted the facts.
Just to accept the fact.
More suggestions(17)
tolerated the fact
recognized the fact
acknowledged the fact
appreciated the fact
accepting the fact
agrees the fact
preferred the fact
recognize the fact
incorporated the fact
accepted the call
accepted the proffer
accepted the outcome
accepted the mission
accepted the app
accepted the charter
accepted the challenge
accepted the responsibility
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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