Description:
This is an interactive problem.
There are $$$n$$$ words in a text editor. The $$$i$$$-th word has length $$$l_i$$$ ($$$1 \leq l_i \leq 2000$$$). The array $$$l$$$ is hidden and only known by the grader.
The text editor displays words in lines, splitting each two words in a line with at least one space. Note that a line does not have to end with a space. Let the height of the text editor refer to the number of lines used. For the given width, the text editor will display words in such a way that the height is minimized.
More formally, suppose that the text editor has width $$$w$$$. Let $$$a$$$ be an array of length $$$k+1$$$ where $$$1=a_1 < a_2 < \ldots < a_{k+1}=n+1$$$. $$$a$$$ is a valid array if for all $$$1 \leq i \leq k$$$, $$$l_{a_i}+1+l_{a_i+1}+1+\ldots+1+l_{a_{i+1}-1} \leq w$$$. Then the height of the text editor is the minimum $$$k$$$ over all valid arrays.
Note that if $$$w < \max(l_i)$$$, the text editor cannot display all the words properly and will crash, and the height of the text editor will be $$$0$$$ instead.
You can ask $$$n+30$$$ queries. In one query, you provide a width $$$w$$$. Then, the grader will return the height $$$h_w$$$ of the text editor when its width is $$$w$$$.
Find the minimum area of the text editor, which is the minimum value of $$$w \cdot h_w$$$ over all $$$w$$$ for which $$$h_w \neq 0$$$.
The lengths are fixed in advance. In other words, the interactor is not adaptive.
Input Format:
The first and only line of input contains a single integer $$$n$$$ ($$$1 \leq n \leq 2000$$$) — the number of words on the text editor.
It is guaranteed that the hidden lengths $$$l_i$$$ satisfy $$$1 \leq l_i \leq 2000$$$.
Output Format:
None
Note:
In the first test case, the words are $$$\{\texttt{glory},\texttt{to},\texttt{ukraine},\texttt{and},\texttt{anton},\texttt{trygub}\}$$$, so $$$l=\{5,2,7,3,5,6\}$$$.
If $$$w=1$$$, then the text editor is not able to display all words properly and will crash. The height of the text editor is $$$h_1=0$$$, so the grader will return $$$0$$$.
If $$$w=9$$$, then a possible way that the words will be displayed on the text editor is:
- $$$\texttt{glory__to}$$$
- $$$\texttt{ukraine__}$$$
- $$$\texttt{and_anton}$$$
- $$$\texttt{__trygub_}$$$
The height of the text editor is $$$h_{9}=4$$$, so the grader will return $$$4$$$.
If $$$w=16$$$, then a possible way that the words will be displayed on the text editor is:
- $$$\texttt{glory_to_ukraine}$$$
- $$$\texttt{and_anton_trygub}$$$
The height of the text editor is $$$h_{16}=2$$$, so the grader will return $$$2$$$.
We have somehow figured out that the minimum area of the text editor is $$$32$$$, so we answer it.