Alec Baldwin 'Has Found Comfort' in His Family as He 'Continues to Grieve' Rust Shooting: Source

Jamie McCarthy/Getty Hilaria and Alec Baldwin

Alec Baldwin is finding comfort in his family after the tragic incident in which he accidentally killed cinematographer Halyna Hutchins on the set of his film Rust.

Baldwin, 63, has left New York City and is spending time with his wife, Hilaria, and their six children: daughters Carmen Gabriela, 8, and María Lucía Victoria, 6 months, and sons Rafael Thomas, 6, Leonardo Ángel Charles, 4, Romeo Alejandro David, 3, and Eduardo "Edu" Pao Lucas, 1.

A source tells PEOPLE Baldwin is "laying low" with family and has "found comfort" with them in the aftermath of the on-set shooting.

"Alec and Hilaria left the city and are laying low with the kids," the source says. "They do go out occasionally to pick up dinner, but are mostly trying to protect the kids' privacy and just spend quiet time together as a family."

Mark Sagliocco/Getty Alec Bal

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The source adds, "Alec continues to grieve the last week's tragic events, but has found comfort being surrounded by his loved ones. He is trying to focus his attention as best he can on being present for his wife and kids as he works through the trauma. Alec continues to cooperate with the sheriff's office in their investigation."

For more on the shooting on the Rust set, listen below to our daily podcast PEOPLE Every Day.

Baldwin had been practicing drawing a prop gun while rehearsing for a scene in Rust on Oct. 21 when the weapon went off, according to an affidavit obtained by PEOPLE. Hutchins, 42, was fatally wounded in the shooting, while director Joel Souza was injured and later released from the hospital.

A preliminary investigation determined that the bullet fired by Baldwin fatally hit Hutchins and then struck Souza, Santa Fe County Sheriff's Office spokesperson Juan Rios told PEOPLE this week.

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Baldwin had been practicing with a prop gun, which was identified as "cold" by Rust assistant director Dave Halls, meaning it did not contain a live round. Halls handed the gun to Baldwin and told him it was "cold," according to a search warrant affidavit from the Santa Fe Sheriff's Office. Halls has not yet issued a public comment addressing the incident.

Baldwin issued a statement on the tragedy Friday morning, the day after the shooting occurred.

"There are no words to convey my shock and sadness regarding the tragic accident that took the life of Halyna Hutchins, a wife, mother and deeply admired colleague of ours," he wrote on Twitter.

"I'm fully cooperating with the police investigation to address how this tragedy occurred and I am in touch with her husband, offering my support to him and his family," Baldwin added. "My heart is broken for her husband, their son, and all who knew and loved Halyna."

The investigation into the Rust incident is currently ongoing, and production on the film has been halted. No charges have been filed at this time.