USACO 2019 January Contest Platinum Problem 3 Train Tracking 2

Every day the express train goes past the farm. It has NN carriages (1≤N≤1051≤N≤105), each with a positive integer label between 11and 109109; different carriages may have the same label.

Usually, Bessie watches the train go by, tracking the carriage labels. But today is too foggy, and Bessie can't see any of the labels! Luckily, she has acquired the sliding window minimums of the sequence of carriage labels, from a reputable source in the city. In particular, she has a positive integer KK, and N−K+1N−K+1 positive integers c1,…,cN+1−Kc1,…,cN+1−K, where cici is the minimum label among carriages i,i+1,…,i+K−1i,i+1,…,i+K−1.

Help Bessie figure out the number of ways to assign a label to each carriage, consistent with the sliding window minimums. Since this number may be very large, Bessie will be satisfied if you find its remainder modulo 109+7109+7.

Bessie's information is completely reliable; that is, it is guaranteed that there is at least one consistent way to assign labels.

OUTPUT FORMAT (file tracking2.out):

A single integer: the number of ways, modulo 109+7109+7, to assign a positive integer not exceeding 109109 to each carriage, such that the minimum label among carriages i,i+1,…,i+K−1i,i+1,…,i+K−1 is cici for each 1≤i≤N−K+11≤i≤N−K+1.

SAMPLE INPUT:

4 2 999999998 999999999 999999998

SAMPLE OUTPUT:

3

Problem credits: Dhruv Rohatgi