## Gondola

Mao-Kong Gondola is a famous attraction in Taipei. The gondola system consists of a circular rail, a single station, and n gondolas numbered consecutively from 1 to n running around the rail in a fixed direction. After gondola i passes the station, the next gondola to pass the station will be gondola i + 1 if i < n, or gondola 1 if i = n.

Gondolas may break down. Luckily we have an infinite supply of spare gondolas, which are numbered n + 1, n + 2, and so on. When a gondola breaks down we replace it (in the same position on the track) with the first available spare gondola, that is, the one with the lowest number. For example, if there are five gondolas and gondola 1 breaks down, then we will replace it with gondola 6.

You like to stand at the station and watch the gondolas as they pass by. A gondola sequence is a sequence of n numbers of gondolas that pass the station. It is possible that one or more gondolas broke down (and were replaced) before you arrived, but none of the gondolas break down while you are watching.

Note that the same configuration of gondolas on the rail can give multiple gondola sequences, depending on which gondola passes first when you arrive at the station. For example, if none of the gondolas have broken down then both (2, 3, 4, 5, 1) and (4, 5, 1, 2, 3) are possible gondola sequences, but (4, 3, 2, 5, 1) is not (because the gondolas appear in the wrong order).

If gondola 1 breaks down, then we might now observe the gondola sequence (4, 5, 6, 2, 3). If gondola 4 breaks down next, we replace it with gondola 7 and we might observe the gondola sequence (6, 2, 3, 7, 5). If gondola 7 breaks down after this, we replace it with gondola 8 and we may now observe the gondola sequence (3, 8, 5, 6, 2).

broken gondolanew gondolapossible gondola sequence
16(4, 5, 6, 2, 3)
47(6, 2, 3, 7, 5)
78(3, 8, 5, 6, 2)

A replacement sequence is a sequence consisting of the numbers of the gondolas that have broken down, in the order in which they break down. In the previous example the replacement sequence is (1, 4, 7). A replacement sequence r produces a gondola sequence g if, after gondolas break down according to the replacement sequence r, the gondola sequence g may be observed.

### Gondola Sequence Checking

In the first three subtasks you must check whether an input sequence is a gondola sequence. See the table below for examples of sequences that are and are not gondola sequences. You need to implement an operation valid.

• valid(n, inputSeq)
• n: the length of the input sequence
• inputSeq: array of length n; inputSeq[i] is element i of the input sequence, for 0 ≤ in − 1.
• For this operation you should output 1 if the input sequence is a gondola sequence, or 0 otherwise.

15n ≤ 100has each number from 1 to n exactly once
25n ≤ 100,0001 ≤ inputSeq[i]n
310n ≤ 100,0001 ≤ inputSeq[i] ≤ 250,000

#### Examples

1(1, 2, 3, 4, 5, 6, 7)1
1(3, 4, 5, 6, 1, 2)1
1(1, 5, 3, 4, 2, 7, 6)01 cannot appear just before 5
1(4, 3, 2, 1)04 cannot appear just before 3
2(1, 2, 3, 4, 5, 6, 5)0two gondolas numbered 5
3(2, 3, 4, 9, 6, 7, 1)1replacement sequence (5, 8)
3(10, 4, 3, 11, 12)04 cannot appear just before 3

### Replacement Sequence

In the next three subtasks you must construct a possible replacement sequence that produces a given gondola sequence. Any such replacement sequence will be accepted. You need to implement an operation replacement.

• replacement(n, gondolaSeq, replacementSeq)
• n is the length of the input sequence
• gondolaSeq: array of length n; gondolaSeq is guaranteed to be a gondola sequence, and gondolaSeq[i] is element i of the sequence, for 0 ≤ in − 1.
• For this operation you should output l, the length of the replacement sequence, followed by l integers representing the replacement sequence.

45n ≤ 1001 ≤ gondolaSeq[i]n + 1
510n ≤ 1,0001 ≤ gondolaSeq[i] ≤ 5,000
620n ≤ 100,0001 ≤ gondolaSeq[i] ≤ 250,000

#### Examples

4(3, 1, 4)1 2(2)
4(5, 1, 2, 3, 4)0( )
5(2, 3, 4, 9, 6, 7, 1)2 5 8(5, 8)

### Count Replacement Sequences

In the next four subtasks you must count the number of possible replacement sequences that produce a given sequence (which may or may not be a gondola sequence), modulo 1,000,000,009. You need to implement an operation countReplacement.

• countReplacement(n, inputSeq)
• n: the length of the input sequence
• inputSeq: array of length n; inputSeq[i] is element i of the input sequence, for 0 ≤ in − 1.
• If the input sequence is a gondola sequence, then count the number of replacement sequences that produce this gondola sequence (which could be extremely large), and output this number modulo 1,000,000,009. If the input sequence is not a gondola sequence, then output 0. If the input sequence is a gondola sequence but no gondolas broke down, then output 1.

#### Subtasks 7, 8, 9, 10

754 ≤ n ≤ 501 ≤ inputSeq[i]n + 3
8154 ≤ n ≤ 501 ≤ inputSeq[i] ≤ 100, and at least n − 3 of the initial
gondolas 1, …, n did not break down.
915n ≤ 100,0001 ≤ inputSeq[i] ≤ 250,000
1010n ≤ 100,0001 ≤ inputSeq[i] ≤ 1,000,000,000

#### Examples

7(1, 2, 7, 6)2(3, 4, 5) or (4, 5, 3)
8(2, 3, 4, 12, 6, 7, 1)1(5, 8, 9, 10, 11)
9(4, 7, 4, 7)0inputSeq is not a gondola sequence
10(3, 4)2(1, 2) or (2, 1)

### Input Format

Line 1: T, the subtask number your program must solve (1 ≤ T ≤ 10).
Line 2: n, the length of the input sequence.
Line 3: If T is 4, 5, or 6, this line contains gondolaSeq[0], …, gondolaSeq[n-1]. Otherwise this line contains inputSeq[0], …, inputSeq[n-1].

### Output Format

If T is 1, 2, or 3: the output should consist of a single integer, 1 if the input sequence is a gondola sequence, or 0 otherwise.
If T is 4, 5, or 6: the output should consist of the integer l, the length of the replacement sequence, followed by l integers on the same line representing the replacement sequence itself.
If T is 7, 8, 9, or 10: the only line of output should consist of a single integer, the number of replacement sequences producing the input sequence modulo 1,000,000,009 if the input sequence is a gondola sequence, 0 if the input sequence is not a gondola sequence, or 1 if the input sequence is a gondola sequence but no gondolas broke down.

### Sample Input 1

1
30
16 26 18 19 20 13 22 21 24 25 17 27 28 29 30 1 2 3 11 5 6 8 7 9 10 12 4 23 14 15

0

2
6
3 4 5 6 1 2

1

3
7
1 2 3 4 5 6 5

0

4
2
3 2

1 1

### Sample Input 5

5
14
12 13 14 1 2 3 4 5 6 7 8 9 10 11

0

### Sample Input 6

6
50
4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 51 47 48 49 50 1 2 3

1 46

### Sample Input 7

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

1

### Sample Input 8

8
14
6 94 8 9 10 93 12 13 95 1 2 3 4 5

224280120

9
4
1 2 7 6

2

10
4
4 7 4 7

### Sample Output 10

0

Point Value: 20 (partial)
Time Limit: 1.00s
Memory Limit: 256M

Languages Allowed:
C++03, PAS, C, HASK, ASM, RUBY, PYTH2, JAVA, PHP, SCM, CAML, PERL, C#, C++11, PYTH3

• (0/0)
WTF i got WA on subtasks 5 and 6. Then I added this to the code:
if (pos[i] < 1 || pos[i] > n)
{
cout << ' ' << pos[i] - 1 << ':';
return 12;
}
if (gondolaSeq[pos[i] - 1] < 0)
{
cout << ' ' << n << ' ' << pos[i] - 1 << '?' << gondolaSeq[pos[i] - 1];
return 11;
}
So something that writes random stuff to stdout in cases that should never happen. Now I get AC

• (0/0)
Does anyone have any hints for speeding up exponentiation modulo a number, or my program's execution in general? I'm achieving a bound of O(n log n), but subtasks 9 and 10 are still giving TLE...

• (1/0)
You are probably using mod too many times, as I see lots of unnecessary mods in your code.

Also make sure you are using fast exponentiation in log k time.

• (0/0)
Amazingly, it was a silly logic error that caused the program to never terminate under certain cases. Thank you for your help, though!

• (0/0)
My solution got AC thought it is wrong ... try this
1 4
1 7 4 9
ans is 0

• (1/0)
"Test data being weak" is a known issue in IOI 2014 :).

• (0/0)
I downloaded test data and my output is same as expected output, but this judge is showing that i am writing bad number at begin.

• (1/0)
I got WA on subtask 5 & 6 again and again.
When I built a sequence as my program printed, I got the same sequence with input data. And this is correct.

• (1/0)
Sorry about that. Hold off on submitting this for now.

• (1/0)
And...?

• (1/0)
Sorry, I've been sick for a few days so I couldn't make a planned change to the back end. It'll get done soon.

• (2/0)
Oh, I'm sorry. I didn't know that. You don't have to hurry. Thanks.

• (0/0)
I've tested in other 2 judges and got AC :)

• (0/0)
The checker for the subtasks should be working properly now. I have rejudged all the WA's.

Note: The length of your replacement sequences should not exceed 250000, or they will be graded as WA.

• (0/0)
thank you :)