JAKARTA, Indonesia — The first officer told the captain that he had been called at 4 a.m. and told to work the flight, which was not on his original schedule. The captain replied that he had the flu. He coughed 15 times in the hour before takeoff.

So began Lion Air Flight 610, which crashed into the Java Sea 13 minutes after taking off from Jakarta, Indonesia’s capital, a year ago, killing all 189 people aboard.

The pilots, one harried and one sick, could not know that they were in an untenable situation: They had been handed a plane that international investigators now say had a fatal design problem.

Their conversation was described in Indonesian investigators’ final report on the crash, which was released on Friday. It blamed a combination of factors for the disaster, including systematic design flaws in the Boeing 737 Max that were compounded by maintenance issues and lapses on the part of the flight crew.