Johnny Solving

Description:

Input:

The first line contains three integers $$$n$$$, $$$m$$$ and $$$k$$$ ($$$1 \leq k \leq n \leq 2.5 \cdot 10^5, 1 \leq m \leq 5 \cdot 10^5$$$)

Next $$$m$$$ lines describe edges of the graph in format $$$v$$$, $$$u$$$ ($$$1 \leq v, u \leq n$$$). It's guaranteed that $$$v \neq u$$$ and all $$$m$$$ pairs are distinct.

It's guaranteed that a degree of each vertex is at least $$$3$$$.

Output:

Print PATH in the first line, if you solve problem for Johnny. In the second line print the number of vertices in the path $$$c$$$ ($$$c \geq \frac{n}{k}$$$). And in the third line print vertices describing the path in route order.

Print CYCLES in the first line, if you solve problem for Solving. In the following lines describe exactly $$$k$$$ cycles in the following format: in the first line print the size of the cycle $$$c$$$ ($$$c \geq 3$$$). In the second line print the cycle in route order. Also, the first vertex in the cycle must be a representative.

Print $$$-1$$$ if there is no any solution. The total amount of printed numbers in the output must be at most $$$10^6$$$. It's guaranteed, that if exists any solution then there is a correct output satisfies this restriction.

Sample Input:

4 6 2
1 2
1 3
1 4
2 3
2 4
3 4

Sample Output:

PATH
4
1 2 3 4 

Sample Input:

10 18 2
1 2
1 3
1 4
1 5
1 6
1 7
1 8
1 9
1 10
2 3
3 4
2 4
5 6
6 7
5 7
8 9
9 10
8 10

Sample Output:

CYCLES
4
4 1 2 3 
4
7 1 5 6