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.

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 $$$l_1,l_2,\ldots,l_n$$$ ($$$1 \leq l_i \leq 2000$$$).