Let $x _ { 1 } , \ldots , x _ { 2024 }$ be non-negative real numbers with $\sum _ { i = 1 } ^ { 2024 } x _ { i } = 1$. Find, with proof, the minimum and maximum possible values of the expression
$$\sum _ { i = 1 } ^ { 1012 } x _ { i } + \sum _ { i = 1013 } ^ { 2024 } x _ { i } ^ { 2 }$$