Finally 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*<sub class="lower-index">1</sub>-th, *x*<sub class="lower-index">2</sub>-th, ..., *x*<sub class="lower-index">*k*</sub>-th panels are in the ON state and other panels are in the OFF state.

She is given an array *a*<sub class="lower-index">1</sub>, ..., *a*<sub class="lower-index">*l*</sub>. In each move, she can perform the following operation: choose an index *i* (1 ≤ *i* ≤ *l*), choose consecutive *a*<sub class="lower-index">*i*</sub> 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.
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*<sub class="lower-index">1</sub>, ..., *x*<sub class="lower-index">*k*</sub> (1 ≤ *x*<sub class="lower-index">1</sub> &lt; *x*<sub class="lower-index">2</sub> &lt; ... &lt; *x*<sub class="lower-index">*k*</sub> ≤ *n*), separated by single spaces.

The third line contains *l* integers *a*<sub class="lower-index">1</sub>, ..., *a*<sub class="lower-index">*l*</sub> (1 ≤ *a*<sub class="lower-index">*i*</sub> ≤ *n*), separated by single spaces. It is possible that some elements of the array *a*<sub class="lower-index">*i*</sub> are equal value.
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.
print("_RANDOM_GUESS_1689184662.115028")# 1689184662.115059