Train Tracking 2 timer 2초 memory 512MB

Every day the express train goes past the farm. It has N carriages (1 \leq N \leq 10^5), each with a positive integer label between 1 and 10^9; 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 K, and N-K+1 positive integers c_1,\dots,c_{N+1-K}, where c_i is the minimum label among carriages i, i+1, \dots, 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 10^9 + 7.

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

Problem credits: Dhruv Rohatgi