Given that $f(x)=|x|+2|x-1|+|x-2|+|x-4|+|x-6|+2|x-10|, x \in(-\infty, \infty)$ the minimum value of $f(x)$ is _________.

If $|x+1| + (y+2)^2 = 0$ and $ax - 3ay = 1$, then the value of $a$ is

If $n$ is an integer such that $\frac{|n+6|-|n-3|}{\sqrt{100-n^3}} \geq 0$, then the number of possible values of $n$ is ___

The area enclosed by the curve $2|x| + 3|y| = 6$ is

The number of real solutions of the equation $x^2 - 10|x| - 56 = 0$ is

The number of pairs $(x, y)$ of integers satisfying the inequality $|x - 5| + |y - 5| \leq 6$ is:

The length of the line segment joining the two intersection points of the curves $y = 4970 - |x|$ and $y = x ^ 2$ is