Gold Stimulation Also Needs a Meal First

Bo-seon went to the Namgang Lantern Festival during a trip to Jinju. The festival has many colorful sculptures and fun shows, but Bo-seon, whose motto is 'Even Mount Geumgang needs a meal first,' went looking for the food truck zone with lots of food. Strangely, there was no food truck zone. It turned out the festival had a cooking zone instead.

The cooking zone sells $N$ kinds of ingredients. Each ingredient has a stimulation level $T_1, \dots, T_N$, and every ingredient is plentiful enough to be used as much as needed. From these ingredients, you choose the kinds of ingredients and how many of each to cook a dish. The stimulation level of a finished dish is the sum of the stimulation levels of the ingredients used. Each cooking session costs a fixed amount of money, and the number of cooking sessions is not limited. However, each dish must use exactly $M$ ingredients. Too many ingredients take too long and ruin the taste, and too few make the taste bland.

Bo-seon plans to eat $K$ dishes. Bo-seon is a gourmet and cares about the stimulation levels of these $K$ dishes. Let $F_i$ be the stimulation level of the $i$-th dish eaten, and let $D_i = F_i - F_1$. The meal plan is the increasing sequence $D = \{D_1, \dots, D_K\}$, where $D_1$ is always $0$. Any dish whose stimulation level differs from $F_1$ by at least $D_1$ and at most $D_K$, and which is not in the plan $D$, cannot be made, because similar dishes can be confused.

For example, let the plan $D$ be $\{0, 2, 3\}$. If $F_1$ is chosen as $3$, then $F_2$ must be $5$ and $F_3$ must be $6$. A dish with stimulation level $4$ is not in the plan, and its difference from $F_1$ is between $D_1$ and $D_K$, so it must not be makeable. If a dish with stimulation level $4$ can be made, $F_1$ cannot be $3$.

Bo-seon makes every dish that can be made using exactly $M$ ingredients. Count the stimulation levels that can serve as $F_1$ while satisfying the conditions above.

The first line contains the number of ingredient kinds $N$, the number of ingredients per dish $M$, and the number of dishes $K$, separated by spaces. $(1 \le N, M \le 1\,000; 1 \le K \le 1\,000\,000)$

The second line contains $T_1, \dots, T_N$, separated by spaces. $T_i$ is the stimulation level of the $i$-th ingredient. $(1 \le T_i \le 1\,000)$

The third line contains $D_1, \dots, D_K$, separated by spaces. This is the increasing sequence $D$ that describes the meal plan. $(0 \le D_i \le 1\,000\,000; D_1 = 0)$

All numbers in the input are integers.

Print one line containing the number of stimulation levels that can be $F_1$ under the conditions.

Python 3 users are recommended to submit with PyPy3.