MAJURO, Marshall Islands (CNN) -- A castaway who claims to have spent more than a year adrift in the Pacific Ocean says his faith in God kept him going through the ordeal. But there were times, he said, when he wanted to end it all.
There were plenty of those bleak moments, he said: When he didn't have food and water. When a boy traveling with him starved to death.
The man, calling himself Jose Salvador Alvarenga, was washed up last week in a heavily damaged boat on a remote coral atoll in the Marshall Islands. He claimed he had been living off fish and turtles he had caught and relying on rainwater, and sometimes his own urine, to drink.
He said he had been lost at sea for 13 months, after setting off from Mexico -- thousands of miles to the east. Many questions remain about how he could have lived on his small boat for so long as it drifted across the ocean.
In an exclusive interview Tuesday, CNN asked Alvarenga how he survived his time at sea.
He pointed upward and said, "God ... My faith in God."
"I thought, 'I am going to get out,'" he said. "Get out, get out, get out."
But he also admitted to dark moments, saying he considered killing himself.
"Twice I wanted to," he said, as he gestured slitting his throat. "I wanted to with a knife. When I didn't have water, food; I gave up and I grabbed a knife."
But he didn't go through with the act, he said, because he was "scared."
During his long period at sea, Alvarenga said he lost track of the date and the day of the week.
He would follow the sun's path across the sky, he said, indicating the movement with his hands as he spoke.
"(I didn't know) the date or the day, only the hours," he said. "Only when it was getting dark and when the light was coming out."
'We will have him back soon'
Authorities are trying to determine the veracity of Alvarenga's story. The Mexican government has confirmed Alvarenga's identity and said he is an El Salvador national who was living in Tonala in Chiapas state.
Julio Camarena Villasenor, the Mexican Ambassador to the Philippines, said Tuesday that Alvarenga is still recovering in a hospital in the Marshall Islands after being found in a "weak state of mind and health."
He'll be repatriated to El Salvador as soon as possible, once he is medically cleared to travel and has the correct documentation, Camarena Villasenor said. No time frame for the repatriation has been established yet.
Alvarenga is from Garita Palmera in El Salvador, where CNN caught up with some members of his family.
They said they hadn't heard from him in about eight years and thought he might be dead. He has a 12-year-old daughter there who doesn't remember her father.
"My heart would tell me that my son was not dead, but I wondered about it so often that I had started to lose faith," said Julia Alvarenga, his mother.
"God willing, my son is not dead. God willing, my son is alive. And we're going to see him again one day. I'm very happy after learning that he's alive and that we will have him back home soon," his father, Ricardo, said.
But seeing what their son has been through also makes brings them feelings of sadness, he added.
"The last time we saw him, we didn't want him to go," his father said.
'Much better shape' than expected
Alvarenga was found on sparsely populated Ebon Atoll, a 22-hour boat ride from the Marshall Islands capital of Majuro, on Thursday.
The southernmost of the Marshall Islands' atolls, Ebon has only 2.2 square miles of land, one phone line and no Internet service. The government airplane that services the atoll was not working, so Alvarenga did not make it to Majuro until Monday morning.
People on the island where he was found Thursday say the 26-foot fiberglass boat was in very bad condition, covered in barnacles and with the carcasses of several turtles littering the deck.
Video from Majuro shows Alvarenga walking a gangplank from a government boat to a waiting ambulance. Waving to those gathered around the dock, he is supported by a medical assistant as he walks. From inside the ambulance, he gives a thumbs up before it drives away.
"He's in much better shape than one would expect after such an ordeal," said Tom Armbruster, U.S. ambassador to the Marshall Islands.
Losing his companion
Alvarenga is said to have set off from the Mexican town of Costa Azul, according to Reynaldo Aguilar Martinez, undersecretary for the state of Chiapas. He claims to have left for what was supposed to be a one-day expedition to catch sharks on December 21, 2012.
Alvarenga said that he and a teenage companion were blown off-course by northerly winds and then caught in a storm, eventually losing use of their engines.
They had no radio signal to report their plight, he said.
According to Anjenette Kattil of the Marshall Islands Ministry of Foreign Affairs, Alvarenga said that four weeks into their drift, he lost the young man because he refused to eat raw birds. There are no details yet on what Alvarenga did with the young man's body.
don't doubt story
If Alvarenga's story proves true, the trip across the Pacific would have taken him across roughly 6,700 miles (10,800 kilometers) of open ocean before ending in the Marshall Islands, about halfway between Hawaii and Australia, in the northern Pacific.
Officials in the Marshall Islands say they have no reason to doubt what he has told them so far, said Camarena Villasenor of Mexico.
Such an amazing journey isn't unheard-of in the small Pacific nation, as three Mexican fishermen made a similar drift voyage in 2006 that lasted nine months. Those men lived off fish they caught and rainwater, and they read the Bible for comfort.
Conditions in the Pacific make the timeline of Alvarenga's journey plausible, according to Judson Jones, a producer for CNN Weather.
Jones said the currents between Mexico and the Marshall Islands would have carried a boat about 27 miles (42 kilometers) a day. That would mean the journey would take about 208 days if the boat stayed in the current. But Jones said a meandering journey in and out of the currents was most likely, making a 13-month journey believable.
Journalists Suzanne Chutaro and Jack Niedenthal reported from Majuro, and CNN's Jethro Mullen wrote from Hong Kong. CNN's Brian Walker, Brad Lendon, Khushbu Shah, Radina Gigova, Rafael Romo, Gustavo Valdes and Nick Parker contributed to this report.
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