The deadly path of the cold-blooded gunman who executed UnitedHealthcare CEO Brian Thompson is being traced by NYPD and FBI investigators in a massive manhunt that entered its third day Friday.

A picture is emerging of a meticulously planned murder plot, with the suspect arriving in New York City more than a week before the shooting outside a Midtown hotel, surveilling the site in advance of the crime and even using a fake ID to check into a youth hostel.

But many gaps remain as the gunman pulled a disappearing act in the nation’s largest city and continues to elude authorities.

Here’s what we know so far:

The suspect got on a Greyhound somewhere between Atlanta and the Big Apple and arrived at the Port Authority Bus Terminal in Midtown on Nov. 24 at around 10 p.m., according to sources and police.

His arrival came 10 full days before he would carry out his assassination plot, according to sources.

Precisely where he boarded the bus remains unknown.

The suspect took a cab from the bus terminal to the the Hilton hotel on Sixth Avenue and West 54th Street, where the ambush was to take place, and staked it out for about a half hour, NYPD Chief of Detectives Joe Kenny said.

At around 11 p.m., he took another cab to the HI New York City Hostel on the Upper West Side, Kenny said.

It’s not known what he did for the next few days. But the suspect checked out from the youth hostel on Nov. 29, but Kenny said police don’t believe he went anywhere, and that this was more of an administrative type of check out.

The suspect checked back into the hostel on Nov. 30, booking a room using a bogus New Jersey ID and paying in cash, sources said.

Cops released surveillance footage from the lobby of the hostel showing the killer flashing a smile at a receptionist who had asked him to remove his mask so she could see his face. 

He obliged the receptionist because he had been flirting with her, sources said.

It’s unclear what the suspect did for the next several days.

Wednesday – the day of the killing – started early for the suspected assassin, according to police and sources.

Kenny said the suspect was seen on surveillance footage around 5:30 a.m. leaving the hostel, on Amsterdam Avenue near West 104th Street, on a bicycle. 

The masked suspect was captured on video around 6:15 a.m. stopping at a Starbucks at West 56th Street and Sixth Avenue, just blocks away from the Hilton, sources said.

He bought coffee, a bottle of water and a Kind bar, and later tossed the coffee cup in a trash can — which cops fished out of the garbage and now have as evidence, according to sources.

Surveillance footage appeared to show the suspect speaking on the phone about 10 minutes before the murder, according to video obtained by the New York Times.

Police have recovered the phone, which sources said was a burner, in a nearby alley, where they also found an Ethos water bottle the suspect is believed to have purchased at Starbucks.

Police said the suspect arrived at the Hilton about five minutes before the shooting.

Thompson, 50, was scheduled to speak at UnitedHealthcare’s annual investor day event at the Hilton at 8 a.m.

He walked alone from his hotel – the Marriott – across the street to the venue around 6:44 a.m., when city streets were still dark, police and sources said.

The shooter, who had been waiting outside the building, began firing at the CEO, who stumbled to the ground after being hit in the back and the leg.

After the assassin discharged several rounds, he fled on foot down an alley before mounting an electric bicycle, which he rode north along Sixth Avenue toward Central Park, where surveillance camera coverage is spotty, police said. 

Authorities received a 911 call about the shooting and first responders arrived shortly afterward to perform CPR on Thompson.

The suspect enters Central Park at 60th Street and rides his bike onto Center Drive at 6:48 a.m., police said.

The suspect emerged from Central Park near West 77th Street, without the distinctive gray backpack he wore in the shooting, sources told The Post.

Kenny said he was spotted at around 6:58 a.m. on West 85th Street and Columbus Avenue, with the bike.

Then at around 7 a.m., he hopped into a cab near West 86th Street and Amsterdam Avenue, and headed north, Kenny added.

The backpack and bicycle have not been recovered, despite an extensive search of Central Park by NYPD officers Wednesday, sources said.

The suspect’s cab arrived at a Port Authority bus center near West 178th Street and Broadway about an hour after the shooting, sources said.

Video footage showed the man enter the bus center but not leave, indicating he may have boarded an out-of-state bus, Kenny said.

— Additional reporting by Chris Nesi