Story highlights Avonte Oquendo's mother was hopeful until the very end, her lawyer says

He says Avonte would be alive if not for "cascade of errors" at school he'd attended

Security video shows the autistic teen leaving a New York City school on October 4

Several of his body parts, some clothes were found last week in the East River

An arm, legs, underwear, dark jeans and size 5½ Air Jordan sneakers turned up on the Queens side of the East River. The jeans and shoes were the same size of Avonte Oquendo, an autistic 14-year-old last seen last fall walking unsupervised out of his school.

Yet his mother was steadfast: "It's not Avonte until it's Avonte."

That time has come.

On Tuesday -- five days after those body parts and scraps of clothing were found -- Vanessa Fontaine learned that DNA tests proved her son's remains indeed had been found.

Her lawyer said that Fontaine has been stoic, strong, focused and hopeful throughout this ordeal. But after police told her the news on Tuesday, "she finally just broke down ... just crying and crying."

"I kept saying, Vanessa, say it again," David Perecman said of his muddled phone conversation. To which she replied through the tears, "It's Avonte, it's Avonte. (The police) came. It's Avonte."

So ended not just her family's search for the teenager, but that of the United States' most populated city. Police deployed sniffer dogs, combed surveillance footage and repeatedly canvassed each of New York City's 468 subway stations because of Avonte's love of trains.

The most poignant, most personal part of the search was Fontaine's recorded voice that was broadcast from patrol cars and other search vehicles. Avonte couldn't communicate verbally and had the mental capacity of a 7- or 8-year-old. But Fontaine hoped that he would hear her and head toward a police car's flashing lights to safety.

"Avonte, this is your mother," the recording said. "You are safe. Walk toward the lights."

Still, while the hunt for Avonte is over, the saga is not.

For one, funeral services for the teen still have to be planned. And admitting he personally is "good and angry," Perecman reeled off a number of accusations against the Riverview School (also known as the Center Boulevard School) in the Long Island City section of Queens.

They included questions such as: Why did it take "several minutes" for Avonte's teacher to notice he wasn't in class? Why did it take 15 minutes for school administrators to find out? Why were police in a neighborhood precinct not alerted for about an hour?

Hinting that the family will sue the city, Perecman said videotapes he's seen suggested "sheer chaos" in a school that he contends is not safe.

"I am convinced, in my heart of hearts, that had a prompt reaction occurred, had some of this cascade of errors not occurred, that the police would have been called, they would have went outside, and they would have found Avonte before this happened," Perecman said.

"He would be home right now. He would be wearing his Air Jordans, and they wouldn't have been found in a river."

Massive hunt for missing teen

A message from CNN left with the school's dean on Tuesday night was not immediately returned. When such allegations were made previously, the school had declined to comment, while the New York City Department of Education had said that it was cooperating with police.

What has been established from official accounts is that at 12:38 p.m. on October 4, surveillance video showed Avonte leaving his school for children with learning disabilities.

Given Avonte's condition, he should have had a monitor at all times. But the video indicated he was alone.

About three hours later, bloodhounds traced his scent to a marsh near his school but lost the trail, a source close to the investigation said. They picked up his scent again an hour later, this time at a nearby subway station.

But while the dogs might have picked up his scent, they couldn't find Avonte.

The search, though, didn't end that day. New York police at one point devoted 50 officers and a task force of detectives to the case, according to then-Police Commissioner Ray Kelly.

The Metropolitan Transit Authority had its workers check each of its facilities each morning. The Department of Environmental Protection brought in special cameras extended on poles to search the sewer system in and around the area where Avonte went missing, the source said.

And Avonte's family pressed their case with the media and the public, even after Kelly said later in October that "unfortunately, we are not hopeful that we're going to find this young man alive."

"My message to my son is that I love him, and we're going to find him," she told CNN's Piers Morgan."You'll come home to your family. And for anyone who has him, please be kind and to let him go."

It wasn't just his family who kept working and maintained hope, Perecman said. So, too, did many New Yorkers who participated in the search or, at the least, kept their eyes out for Avonte, helped by the fliers and posters that blanketed the city.

Said Perecman: "I don't remember a search in the city of New York that rose to this level. It was a beautiful sight."

Body parts, clothes found in East River

Whatever the outcome, Avonte's family laid the groundwork to sue New York's government days after he went missing. They filed a claim -- the first step in a lawsuit -- over what they described as "the negligence, carelessness and/or recklessness" of those who'd been charged with watching Avonte that day.

Then came last week's discovery of clothes that matched those worn by the slim, approximately 5-foot, 3-inch teenager, as well as the remains.

The fact they were found in the East River also jibed with a possible storyline, given the Riverview School's location near the water around where the Queens Midtown Tunnel goes to and from Manhattan.

The matching clothes were not enough for Fontaine and company. "They weren't going to accept anything else but DNA," her lawyer said. And DNA, ultimately, proved that the remains indeed belonged to Avonte.

So how and when did Avonte die?

The medical examiner hasn't issued a public ruling, though the gory state of the teenager's remains leaves open an alarming array of possibilities. Perecman said he doesn't know if foul play is to blame, whether the fact the boy was found in pieces stems from natural decomposition in the water or something more gruesome.

"You know what they say, you put 10 people in a room you get 12 opinions," the lawyer said of the diverse range of theories as to what happened.