Good evening, contestants.

If a and d are relatively prime positive integers, the arithmetic sequence beginning with a and increasing by d, i.e., a, a + d, a + 2d, a + 3d, a + 4d, ..., contains infinitely many prime numbers. This fact is known as Dirichlet's Theorem on Arithmetic Progressions, which had been conjectured by Johann Carl Friedrich Gauss (1777 - 1855) and was proved by Johann Peter Gustav Lejeune Dirichlet (1805 - 1859) in 1837.

For example, the arithmetic sequence beginning with 2 and increasing by 3, i.e.,

2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35, 38, 41, 44, 47, 50, 53, 56, 59, 62, 65, 68, 71, 74, 77, 80, 83, 86, 89, 92, 95, 98, ... ,

contains infinitely many prime numbers

2, 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, ... .

Your mission, should you decide to accept it, is to write a program to find the nth prime number in this arithmetic sequence for given positive integers a, d, and n.

As always, should you or any of your team be tired or confused, the secretary disavow any knowledge of your actions. This judge system will self-terminate in three hours. Good luck!

Input

Output

Sample Input

Output for the Sample Input

Organize Your Train part II

RJ Freight, a Japanese railroad company for freight operations has recently constructed exchange lines at Hazawa, Yokohama. The layout of the lines is shown in Figure B-1.

Figure B-1: Layout of the exchange lines

A freight train consists of 2 to 72 freight cars. There are 26 types of freight cars, which are denoted by 26 lowercase letters from "a" to "z". The cars of the same type are indistinguishable from each other, and each car's direction doesn't matter either. Thus, a string of lowercase letters of length 2 to 72 is sufficient to completely express the configuration of a train.

Upon arrival at the exchange lines, a train is divided into two sub-trains at an arbitrary position (prior to entering the storage lines). Each of the sub-trains may have its direction reversed (using the reversal line). Finally, the two sub-trains are connected in either order to form the final configuration. Note that the reversal operation is optional for each of the sub-trains.

For example, if the arrival configuration is "abcd", the train is split into two sub-trains of either 3:1, 2:2 or 1:3 cars. For each of the splitting, possible final configurations are as follows ("+" indicates final concatenation position):

  [3:1]
    abc+d  cba+d  d+abc  d+cba
  [2:2]
    ab+cd  ab+dc  ba+cd  ba+dc  cd+ab  cd+ba  dc+ab  dc+ba
  [1:3]
    a+bcd  a+dcb  bcd+a  dcb+a

Excluding duplicates, 12 distinct configurations are possible.

Given an arrival configuration, answer the number of distinct configurations which can be constructed using the exchange lines described above.

Input

Output

Sample Input

Output for the Sample Input

Hexerpents of Hexwamp

Hexwamp is a strange swamp, paved with regular hexagonal dimples. Hexerpents crawling in this area are serpents adapted to the environment, consisting of a chain of regular hexagonal sections. Each section fits in one dimple.

Hexerpents crawl moving some of their sections from the dimples they are in to adjacent ones. To avoid breaking their bodies, sections that are adjacent to each other before the move should also be adjacent after the move. When one section moves, sections adjacent to it support the move, and thus they cannot move at that time. Any number of sections, as far as no two of them are adjacent to each other, can move at the same time.

You can easily find that a hexerpent can move its sections at its either end to only up to two dimples, and can move intermediate sections to only one dimple, if any.

For example, without any obstacles, a hexerpent can crawl forward twisting its body as shown in Figure C-1, left to right. In this figure, the serpent moves four of its eight sections at a time, and moves its body forward by one dimple unit after four steps of moves. Actually, they are much better in crawling sideways, like sidewinders.

Figure C-1: Crawling forward

Their skin is so sticky that if two sections of a serpent that are not originally adjacent come to adjacent dimples (Figure C-2), they will stick together and the serpent cannot but die. Two sections cannot fit in one dimple, of course. This restricts serpents' moves further. Sometimes, they have to make some efforts to get a food piece even when it is in the dimple next to their head.

Figure C-2: Fatal case

Hexwamp has rocks here and there. Each rock fits in a dimple. Hexerpents' skin does not stick to rocks, but they cannot crawl over the rocks. Although avoiding dimples with rocks restricts their moves, they know the geography so well that they can plan the fastest paths.

You are appointed to take the responsibility of the head of the scientist team to carry out academic research on this swamp and the serpents. You are expected to accomplish the research, but never at the sacrifice of any casualty. Your task now is to estimate how soon a man-eating hexerpent may move its head (the first section) to the position of a scientist in the swamp. Their body sections except for the head are quite harmless and the scientist wearing high-tech anti-sticking suit can stay in the same dimple with a body section of the hexerpent.

Input

The input is a sequence of several datasets, and the end of the input is indicated by a line containing a single zero. The number of datasets never exceeds 10.

Each dataset looks like the following.

the number of sections the serpent has (=n)
x₁ y₁
x₂ y₂
...
x_n y_n
the number of rocks the swamp has (=k)
u₁ v₁
u₂ v₂
...
u_k v_k
X Y

The first line of the dataset has an integer n that indicates the number of sections the hexerpent has, which is 2 or greater and never exceeds 8. Each of the n following lines contains two integers x and y that indicate the coordinates of a serpent's section. The lines show the initial positions of the sections from the serpent's head to its tail, in this order.

The next line of the dataset indicates the number of rocks k the swamp has, which is a non-negative integer not exceeding 100. Each of the k following lines contains two integers u and v that indicate the position of a rock.

Finally comes a line containing two integers X and Y, indicating the goal position of the hexerpent, where the scientist is. The serpent's head is not initially here.

All of the coordinates x, y, u, v, X, and Y are between −999999 and 999999, inclusive. Two integers in a line are separated by a single space. No characters other than decimal digits, minus signs, and spaces to separate two integers appear in the input. The coordinate system used to indicate a position is as shown in Figure C-3.

Figure C-3: The coordinate system

Output

Sample Input

Output for the Sample Input

Curling 2.0

On Planet MM-21, after their Olympic games this year, curling is getting popular. But the rules are somewhat different from ours. The game is played on an ice game board on which a square mesh is marked. They use only a single stone. The purpose of the game is to lead the stone from the start to the goal with the minimum number of moves.

Fig. D-1 shows an example of a game board. Some squares may be occupied with blocks. There are two special squares namely the start and the goal, which are not occupied with blocks. (These two squares are distinct.) Once the stone begins to move, it will proceed until it hits a block. In order to bring the stone to the goal, you may have to stop the stone by hitting it against a block, and throw again.

Fig. D-1: Example of board (S: start, G: goal)

The movement of the stone obeys the following rules:

At the beginning, the stone stands still at the start square.
The movements of the stone are restricted to x and y directions. Diagonal moves are prohibited.
When the stone stands still, you can make it moving by throwing it. You may throw it to any direction unless it is blocked immediately(Fig. D-2(a)).
Once thrown, the stone keeps moving to the same direction until one of the following occurs:
- The stone hits a block (Fig. D-2(b), (c)).
  - The stone stops at the square next to the block it hit.
  - The block disappears.
- The stone gets out of the board.
  - The game ends in failure.
- The stone reaches the goal square.
  - The stone stops there and the game ends in success.
You cannot throw the stone more than 10 times in a game. If the stone does not reach the goal in 10 moves, the game ends in failure.

Fig. D-2: Stone movements

Under the rules, we would like to know whether the stone at the start can reach the goal and, if yes, the minimum number of moves required.

With the initial configuration shown in Fig. D-1, 4 moves are required to bring the stone from the start to the goal. The route is shown in Fig. D-3(a). Notice when the stone reaches the goal, the board configuration has changed as in Fig. D-3(b).

Fig. D-3: The solution for Fig. D-1 and the final board configuration

Input

The input is a sequence of datasets. The end of the input is indicated by a line containing two zeros separated by a space. The number of datasets never exceeds 100.

Each dataset is formatted as follows.

the width(=w) and the height(=h) of the board
First row of the board
...
h-th row of the board

The width and the height of the board satisfy: 2 <= w <= 20, 1 <= h <= 20.
Each line consists of w decimal numbers delimited by a space. The number describes the status of the corresponding square.

0 vacant square

1 block

2 start position

3 goal position

The dataset for Fig. D-1 is as follows:

6 6
1 0 0 2 1 0
1 1 0 0 0 0
0 0 0 0 0 3
0 0 0 0 0 0
1 0 0 0 0 1
0 1 1 1 1 1

Output

Sample Input

Output for the Sample Input

The Genome Database of All Space Life

In 2300, the Life Science Division of Federal Republic of Space starts a very ambitious project to complete the genome sequencing of all living creatures in the entire universe and develop the genomic database of all space life. Thanks to scientific research over many years, it has been known that the genome of any species consists of at most 26 kinds of molecules, denoted by English capital letters (i.e. A to Z).

What will be stored into the database are plain strings consisting of English capital letters. In general, however, the genome sequences of space life include frequent repetitions and can be awfully long. So, for efficient utilization of storage, we compress N-times repetitions of a letter sequence seq into N(seq), where N is a natural number greater than or equal to two and the length of seq is at least one. When seq consists of just one letter c, we may omit parentheses and write Nc.

For example, a fragment of a genome sequence:

ABABABABXYXYXYABABABABXYXYXYCCCCCCCCCC

can be compressed into:

4(AB)XYXYXYABABABABXYXYXYCCCCCCCCCC

by replacing the first occurrence of ABABABAB with its compressed form. Similarly, by replacing the following repetitions of XY, AB, and C, we get:

4(AB)3(XY)4(AB)3(XY)10C

Since C is a single letter, parentheses are omitted in this compressed representation. Finally, we have:

2(4(AB)3(XY))10C

by compressing the repetitions of 4(AB)3(XY). As you may notice from this example, parentheses can be nested.

Your mission is to write a program that uncompress compressed genome sequences.

Input

The input consists of multiple lines, each of which contains a character string s and an integer i separated by a single space.

The character string s, in the aforementioned manner, represents a genome sequence. You may assume that the length of s is between 1 and 100, inclusive. However, of course, the genome sequence represented by s may be much, much, and much longer than 100. You may also assume that each natural number in s representing the number of repetitions is at most 1,000.

The integer i is at least zero and at most one million.

A line containing two zeros separated by a space follows the last input line and indicates the end of the input.

Output

Sample Input

Output for the Sample Input

Secrets in Shadows

Long long ago, there were several identical columns (or cylinders) built vertically in a big open space near Yokohama (Fig. F-1). In the daytime, the shadows of the columns were moving on the ground as the sun moves in the sky. Each column was very tall so that its shadow was very long. The top view of the shadows is shown in Fig. F-2.

The directions of the sun that minimizes and maximizes the widths of the shadows of the columns were said to give the important keys to the secrets of ancient treasures.

Fig. F-1: Columns (or cylinders)

Fig. F-2: Top view of the columns (Fig. F-1) and their shadows

The width of the shadow of each column is the same as the diameter of the base disk. But the width of the whole shadow (the union of the shadows of all the columns) alters according to the direction of the sun since the shadows of some columns may overlap those of other columns.

Fig. F-3 shows the direction of the sun that minimizes the width of the whole shadow for the arrangement of columns in Fig. F-2.

Fig. F-3: The direction of the sun for the minimal width of the whole shadow

Fig. F-4 shows the direction of the sun that maximizes the width of the whole shadow. When the whole shadow is separated into several parts (two parts in this case), the width of the whole shadow is defined as the sum of the widths of the parts.

Fig. F-4: The direction of the sun for the maximal width of the whole shadow

A direction of the sun is specified by an angle θ defined in Fig. F-5. For example, the east is indicated by θ=0, the south by θ=π/2, and the west by θ=π. You may assume that the sun rises in the east (θ=0) and sets in the west (θ=π).

Your job is to write a program that, given an arrangement of columns, computes two directions θ_min and θ_max of the sun that give the minimal width and the maximal width of the whole shadow, respectively.

The position of the center of the base disk of each column is specified by its (x,y) coordinates. The x-axis and y-axis are parallel to the line between the east and the west and that between the north and the south, respectively. Their positive directions indicate the east and the north, respectively.

You can assume that the big open space is a plane surface.

Fig. F-5: The definition of the angle of the direction of the sun

There may be more than one θ_min or θ_max for some arrangements in general, but here, you may assume that we only consider the arrangements that have unique θ_min and θ_max in the range 0<=θ_min<π, 0<=θ_max<π.

Input

The input consists of multiple datasets, followed by the last line containing a single zero.

Each dataset is formatted as follows.

n
x₁ y₁
x₂ y₂
...
x_n y_n

n is the number of the columns in the big open space. It is a positive integer no more than 100.

x_k and y_k are the values of x-coordinate and y-coordinate of the center of the base disk of the k-th column (k=1, ..., n). They are positive integers no more than 30. They are separated by a space.

Note that the radius of the base disk of each column is one unit (the diameter is two units). You may assume that some columns may touch each other but no columns overlap others.

For example, a dataset

3
1 1
3 1
4 3

corresponds to the arrangement of three columns depicted in Fig. F-6. Two of them touch each other.

Fig. F-6: An arrangement of three columns

Output

For each dataset in the input, two lines should be output as specified below. The output lines should not contain extra characters such as spaces.

In the first line, the angle θ_min, the direction of the sun giving the minimal width, should be printed. In the second line, the other angle θ_max, the direction of the sun giving the maximal width, should be printed.

Each angle should be contained in the interval between 0 and π (abbreviated to [0, π]) and should not have an error greater than ε=0.0000000001 (=10^-10).

When the correct angle θ is in [0,ε], approximate values in [0,θ+ε] or in [π+θ-ε, π] are accepted. When the correct angle θ is in [π-ε, π], approximate values in [0, θ+ε-π] or in [θ-ε, π] are accepted.

You may output any number of digits after the decimal point, provided that the above accuracy condition is satisfied.