CF1672E.notepad.exe

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.

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 $1 \leq l_i \leq 2000$ .

Begin the interaction by reading $n$ .

To make a query, print "? $w$ " (without quotes, $1 \leq w \leq 10^9$ ). Then you should read our response from standard input, that is, $h_w$ .

If your program has made an invalid query or has run out of tries, the interactor will terminate immediately and your program will get a verdict Wrong answer.

To give the final answer, print "! $area$ " (without the quotes). Note that giving this answer is not counted towards the limit of $n+30$ queries.

After printing a query do not forget to output end of line and flush the output. Otherwise, you will get Idleness limit exceeded. To do this, use:

• fflush(stdout) or cout.flush() in C++;
• System.out.flush() in Java;
• flush(output) in Pascal;
• stdout.flush() in Python;
• see documentation for other languages.

Hacks

The first line of input must contain a single integer $n$ ( $1 \leq n \leq 2000$ ) — the number of words in the text editor.

The second line of input must contain exactly $n$ space-separated integers $l_1,l_2,\ldots,l_n$ ( $1 \leq l_i \leq 2000$ ).

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.