# Problem G

Gyrating Glyphs

You are rocking the latest breakthrough in Computer Science:
animated fonts. Suddenly, all of your colleagues’ code looks
amazing, and you are finally motivated to review it.
Unfortunately, due to the constant rotations, it is hard to
distinguish between the $+$ (plus) and the $\times $ (multiply) operators (all
the other characters are still readable). The function you are
reviewing takes as input $n+1$ integers $a_0, a_1, \ldots , a_ n$ and returns
the value

where the $n$ operators $\operatorname {op}_1,\, \operatorname {op}_2,\, \ldots ,\, \operatorname {op}_ n$ are either $+$ or $\times $. For example when given input $(a_0,a_1,a_2) = (1,1,2)$ with hidden operators $(\operatorname {op}_1,\operatorname {op}_2)=(+,\times )$, then the function returns $((1+1)\times 2)=4 \bmod 10^9+7$.

You can still execute the function a few times on some input and read the returned value. Use this to recover the operators.

## Interaction

This is an interactive problem. Your submission will be run
against an *interactor*, which reads the standard output
of your submission and writes to the standard input of your
submission. This interaction needs to follow a specific
protocol:

The interactor first sends one line containing one integer $n$ ($1 \leq n \leq 20\, 000$), the number of hidden operators.

Then, your program should make at most $1400$ queries to determine the
operators. Each query is made by printing one line of the form
“`?` $a_0\ a_1\ \ldots \ a_ n$”
($0 \leq a_ i <
10^9+7$). The interactor will respond by printing one
line with an integer, the value of

Make sure you flush the buffer after each write.

When you have determined the operators, print a single line
of the form “`!` $s$”, where $s$ is a string consisting of exactly
$n$ characters, which are
all “`+`” (plus) or “`x`” (multiply; this is the lowercase letter
“`x`”, not the Unicode “$\times $” symbol). The $i$th character of this string should
be $\operatorname {op}_
i$. This line does not count as one of your queries.

Using more than $1400$
queries will result in a wrong answer verdict.

A testing tool is provided to help you develop your
solution.

Read | Sample Interaction 1 | Write |
---|

2

? 1 1 2

4

? 1 1 3

6

! +x

Read | Sample Interaction 2 | Write |
---|

10

? 1 1 1 1 1 1 1 1 1 1 1

5

? 0 4 2 4 2 4 2 4 2 4 2

6224

? 1 2 3 4 5 6 7 8 9 10 11

640750

! ++xxx+x+xx