F. Deranged Calculator
time limit per test
2 seconds
memory limit per test
256 megabytes
input
standard input
output
standard output

Hacks are disabled on this problem.

We are excited to introduce a new language for this Codeforces round: the Deranged Calculator language (DC).

The DC language is a simplified programming language with some weird constraints. It only supports a single expression on one line and only has a single positive integer $$$n$$$ as input. The syntax of the expression is similar to that of most programming languages (in fact, the expression is valid Python), but it is very limited:

  • Expressions consist of the input parameter $$$n$$$, parenthesized expressions, and calls to the round function. These are joined together by the operators +, -, *, and /.
  • Grouping with parentheses () can be used to change the order of operations, as usual.
  • The operators * and / have equal precedence and take precedence over + and -, which also have equal precedence. Operators of equal precedence are evaluated from left to right.
  • The round function takes a single expression as its argument and rounds its value to the nearest integer, rounding up when the fractional part is exactly $$$0.5$$$. (For example, round(n/(n-n-n-n)) will always evaluate to $$$0$$$, because $$$x/(x-x-x-x)=-\frac{1}{2}$$$ for a positive integer $$$x$$$. The fractional part is $$$0.5$$$, so it will be rounded up to $$$0$$$.)
  • Only the input value $$$n$$$ can be used; no numeric constants are allowed in the program.
  • All operations are performed using exact fractional arithmetic, so division does not round and can produce fractions.

A derangement of $$$n$$$ numbers is a permutation of those numbers in which none of the numbers appears in its original position. For example, the derangements of the sequence $$$[1, 2, 3]$$$ are $$$[2, 3, 1]$$$ and $$$[3, 1, 2]$$$.

You are given a single integer $$$k$$$. Your job is to make a valid DC program that, for every integer $$$n$$$ ($$$2 \le n \le k$$$), calculates the number of derangements of the sequence $$$[1, 2, \ldots, n]$$$, when $$$n$$$ is given as input to the DC program. The length of the program should not exceed $$$10\,000$$$ characters.

A local testing tool is provided to help you develop your solution. It can be found under Contest Materials.

Input

The only line of input contains a single integer $$$k$$$ ($$$k \in \{2,3,50\}$$$).

Your DC program should output the correct number of derangements for all integers $$$n$$$ such that $$$2 \le n \le k$$$.

Output

Output a valid DC expression on a single line. The expression should only consist of the characters round()+-*/ and the length of the expression should not exceed $$$10\,000$$$ characters.

Your expression will be run on all integers $$$n$$$ such that $$$2 \le n \le k$$$. For each such run, the DC expression should not divide by $$$0$$$ at any time, and its value must be the number of derangements of $$$n$$$.

Examples
Input
2
Output
n-n/n
Input
3
Output
n+round(n-n*(n/(n+n)))-n
Note

For the first example, n-n/n is a valid DC program that calculates $$$n-1$$$. The number of derangements for $$$n=2$$$ is $$$1$$$, so this program correctly computes the answer for $$$n=2$$$.

In the second example, a slightly overcomplicated expression is used, which shows all the features of the language in action. For $$$n=3$$$, the value of the expression is: $$$3+\text{round}(3-3 \cdot(\frac{3}{3+3}))-3 = \text{round}(3 - \frac{3}{2}) = \text{round}(\frac{3}{2}) = \text{round}(1.5)=2$$$. It correctly calculates the number of derangements of the sequence $$$[1,2,3]$$$. For $$$n=2$$$ it can be verified that the value of the expression becomes $$$1$$$.