1) $\left| x-1 \right| + \left| x-2 \right| +\cdots + \left| x-2023 \right| + \left| x-2024 \right|$ Find the minimum value of the expression and when is it minimized.

2) Find positive number $a$ such that $\left| x+1 \right| + \left| x-6 \right| + 2\left| x-a \right|$ has a minimum value of 8.

3) Given that $\left| x-1 \right| + 8\left| x-2 \right| + a\left| x-3 \right| + 2\left| x-4 \right|$ has a minimum value of 12, find the range of $a$.