2 Samuel 1 tells Saul's death differently. Which account is right?

Here are both, as each book gives them. 1 Samuel 31 tells it in the narrator's own voice: Saul asked his armor bearer to kill him, the armor bearer would not, and Saul fell on his own sword (31:4-5). In 2 Samuel 1, while David is at Ziklag (2 Samuel 1:1), a man comes to him out of the camp from Saul (2 Samuel 1:2) and says, "I am an Amalekite" (2 Samuel 1:8). He says that he found Saul on Mount Gilboa "leaning on his spear" (2 Samuel 1:6), and "So I stood beside him and killed him, because I was sure that he could not live after he had fallen" (2 Samuel 1:10), and that he has brought David "the crown that was on his head and the bracelet that was on his arm" (2 Samuel 1:10). 2 Samuel doesn't say whether his account is true. It tells what David said to him: "Why were you not afraid to stretch out your hand to destroy the LORD’s anointed?" (2 Samuel 1:14), and "your mouth has testified against you" (2 Samuel 1:16). The two books set the accounts down without explaining how they fit together, and this answer doesn't go past them.