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D. Password

time limit per test

1 secondmemory limit per test

256 megabytesinput

standard inputoutput

standard outputFinally Fox Ciel arrived in front of her castle!

She have to type a password to enter her castle. An input device attached to her castle is a bit unusual.

The input device is a 1 × *n* rectangle divided into *n* square panels. They are numbered 1 to *n* from left to right. Each panel has a state either ON or OFF. Initially all panels are in the OFF state. She can enter her castle if and only if *x*_{1}-th, *x*_{2}-th, ..., *x*_{k}-th panels are in the ON state and other panels are in the OFF state.

She is given an array *a*_{1}, ..., *a*_{l}. In each move, she can perform the following operation: choose an index *i* (1 ≤ *i* ≤ *l*), choose consecutive *a*_{i} panels, and flip the states of those panels (i.e. ON → OFF, OFF → ON).

Unfortunately she forgets how to type the password with only above operations. Determine the minimal number of operations required to enter her castle.

Input

The first line contains three integers *n*, *k* and *l* (1 ≤ *n* ≤ 10000, 1 ≤ *k* ≤ 10, 1 ≤ *l* ≤ 100), separated by single spaces.

The second line contains *k* integers *x*_{1}, ..., *x*_{k} (1 ≤ *x*_{1} < *x*_{2} < ... < *x*_{k} ≤ *n*), separated by single spaces.

The third line contains *l* integers *a*_{1}, ..., *a*_{l} (1 ≤ *a*_{i} ≤ *n*), separated by single spaces. It is possible that some elements of the array *a*_{i} are equal value.

Output

Print the minimal number of moves required to type the password. If it's impossible, print -1.

Examples

Input

10 8 2

1 2 3 5 6 7 8 9

3 5

Output

2

Input

3 2 1

1 2

3

Output

-1

Note

One possible way to type the password in the first example is following: In the first move, choose 1st, 2nd, 3rd panels and flip those panels. In the second move, choose 5th, 6th, 7th, 8th, 9th panels and flip those panels.

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