# Food Carts

Peter is in charge of food service on a passenger train. The train has $n$ cars ordered sequentially starting from the locomotive. The $i$-th car has $p_ i$ passengers. There are $m$ food carts Peter may put into service. Each food cart serves a unique type of food. The $j$-th food cart can move between the $l_ j$-th car and the $r_ j$-th car (both inclusive) and serve the passengers in these cars.

The passengers on the train are not only hungry, but also
adventurous: as long as a food cart is in service, they would
love to try it! A passenger will be *happy* if
he/she sits in a car that is served by all the food carts that
are in service. Peter would like to design service plans that
run one or more of the $m$
food carts so that at least $k$ of the passengers on the train are
happy.

Peter wants to count the number of different service plans he may choose from (modulo $10^9 + 7$). Two service plans are different if one food cart is put into service in one plan but not in the other plan.

## Input

The first line has three integers $n$, $m$, and $k$ ($1 \leq n, m \leq 2 \cdot 10^5, 1 \leq k \leq 10^{14}$). The next line has $n$ integers. The $i$-th integer is $p_ i$ ($1 \leq p_ i \leq 10^9 $), the number of passengers in the $i$-th car. Each of the next $m$ lines describes a food cart. The $j$-th of these lines has two integers $l_ j$ and $r_ j$ ($1 \leq l_ j \leq r_ j \leq n$), giving the service range of the $j$-th food cart.

## Output

Output the number of different service plans, modulo $10^9 + 7$.

Sample Input 1 | Sample Output 1 |
---|---|

4 3 3 1 2 3 4 1 2 2 3 3 4 |
4 |

Sample Input 2 | Sample Output 2 |
---|---|

5 4 6 1 2 3 4 5 1 4 2 3 3 5 4 5 |
5 |

Sample Input 3 | Sample Output 3 |
---|---|

4 4 1 1 2 3 4 1 4 1 4 1 4 1 4 |
15 |