Amin records the price of his stock every now and then as a data point $$$(t_i, p_i)$$$ in his notebook, where $$$p_i$$$ is the price of his stock at day $$$t_i$$$. The sequence of these data points represents a piecewise-linear function $$$F$$$ displaying the history of prices over a period of time. Indeed, $$$F$$$ connects every pair of consecutive points by a straight line segment. If the price is not recorded for some day $$$t$$$, $$$F(t)$$$ can be used as an estimate instead.

His collected data is getting larger and larger as he has been tracking the price of his stock over a long period of time. Therefore, he has decided to reduce his data by throwing away some of his recoded data points and constructing a new piecewise-linear function $$$F'$$$ with the remaining points. $$$F'$$$ is a so-called "simplification" of $$$F$$$. Let $$$F$$$ be defined over a strictly increasing sequence $$$L = \langle{t_1, \dots, t_n}\rangle$$$ of days, and $$$F'$$$ be defined over a subsequence $$$L' = \langle{t'_1, \dots, t'_m}\rangle$$$ of $$$L$$$, where $$$t'_1=t_1$$$, $$$t'_m=t_n$$$, and $$$F'(t'_i) = F(t'_i)$$$ for $$$1 \leq i \leq m$$$. (We call $$$m$$$ the size of $$$F'$$$.)

For example, in the following figure, we have $$$6$$$ data points, labeled $$$1$$$ through $$$6$$$, whose coordinates are the same as those presented in the second sample input, and $$$F'$$$ is a simplification of $$$F$$$ of size $$$3$$$ with data points $$$1$$$, $$$4$$$ and $$$6$$$. In this figure, $$$F$$$ is depicted by solid lines, and $$$F'$$$ by dashed lines. The error measure for $$$F'$$$ is realized by the vertical distance of point $$$2$$$ to $$$F'$$$.

Amin's goal is to minimize the size of $$$F'$$$, while the error of $$$F'$$$ is bounded by a given value $$$\delta$$$.