Graphic Settings
Contest: Educational Codeforces Round 29
Difficulty rating: 3200

Recently Ivan bought a new computer. Excited, he unpacked it and installed his favourite game. With his old computer Ivan had to choose the worst possible graphic settings (because otherwise the framerate would be really low), but now he wants to check, maybe his new computer can perform well even with the best possible graphics?

There are m graphics parameters in the game. i-th parameter can be set to any positive integer from 1 to ai, and initially is set to bi (bi ≤ ai). So there are $$p = \prod_{i=1}^{m} a_i$$ different combinations of parameters. Ivan can increase or decrease any of these parameters by 1; after that the game will be restarted with new parameters (and Ivan will have the opportunity to check chosen combination of parameters).

Ivan wants to try all p possible combinations. Also he wants to return to the initial settings after trying all combinations, because he thinks that initial settings can be somehow best suited for his hardware. But Ivan doesn't really want to make a lot of restarts.

So he wants you to tell the following:

- If there exists a way to make exactly p changes (each change either decreases or increases some parameter by 1) to try all possible combinations and return to initial combination, then Ivan wants to know this way.
- Otherwise, if there exists a way to make exactly p - 1 changes to try all possible combinations (including the initial one), then Ivan wants to know this way.

Help Ivan by showing him the way to change parameters!

Input format:
The first line of input contains one integer number m (1 ≤ m ≤ 6).

The second line contains m integer numbers a1, a2, ..., am (2 ≤ ai ≤ 1000). It is guaranteed that $$p = \prod_{i=1}^{m} a_i \leq 10^5$$.

The third line contains m integer numbers b1, b2, ..., bm (1 ≤ bi ≤ ai).

Output format:
If there is a way to make exactly p changes (each change either decreases or increases some parameter by 1) to try all possible combinations and return to initial combination, then output Cycle in the first line. Then p lines must follow, each desribing a change. The line must be either inc x (increase parameter x by 1) or dec x (decrease it).

Otherwise, if there is a way to make exactly p - 1 changes to try all possible combinations (including the initial one), then output Path in the first line. Then p - 1 lines must follow, each describing the change the same way as mentioned above.

Otherwise, output No.

Constraints:
Time limit: 2.0 s
Memory limit: 256.0 MB

Example input:
1
3
1

Example output:
Path
inc 1
inc 1

Example input:
1
3
2

Example output:
No

Editorial:
If m = 1, then everything is simple. Cycle exists if a[1] = 2, and path exists if b[1] = 1 or b[1] = a[1].
Let's consider the case when m = 2. Let's call the combination odd if the sum of parameters is odd for this combination, and even otherwise. It's easy to see that if a[1] and a[2] are both odd, then it's impossible to construct a cycle because the number of even combinations is greater than the number of odd combinations. Furthermore, it's impossible to construct a path if our initial combination is an odd one.
Let's show how to construct answer in all other cases. Constructing a cycle is easy if at least one of a[i] is even:
And constructing a path can be divided into four cases.
Starting point in corner:
Start near the border:
Both coordinates of starting point are even:
Both coordinates are odd:
So the case when m = 2 is solved.
Now, if m = 3, then it can be reduced to m = 2 as follows:
Suppose we have a[1] = 3, a[2] = 3, a[3] = 3. Then let's at first consider only combinations where the third parameter is equal to 1:
Then, if we need to add the combinations with third parameter equal to 2, we mirror the "layer":
And if we need to add the third "layer", we mirror it again:
And so on.
This way can also be used to reduce m > 3 to m = 2.