Chiaki has a sequence A={a1,a2,…,an}. Let RMQ(A,l,r) be the minimum i (l≤i≤r) such that ai is the maximum value in al,al+1,…,ar.
Two sequences A and B are called \textit{RMQ Similar}, if they have the same length n and for every 1≤l≤r≤n, RMQ(A,l,r)=RMQ(B,l,r).
For a given the sequence A={a1,a2,…,an}, define the weight of a sequence B={b1,b2,…,bn} be ∑i=1nbi (i.e. the sum of all elements in B) if sequence B and sequence A are RMQ Similar, or 0 otherwise. If each element of B is a real number chosen independently and uniformly at random between 0 and 1, find the expected weight of B.
Input
There are multiple test cases. The first line of input contains an integer T, indicating the number of test cases. For each test case: The first line contains an integer n (1≤n≤106) -- the length of the sequence. The second line contains n integers a1,a2,…,an (1≤ai≤n) denoting the sequence. It is guaranteed that the sum of all n does not exceed 3×106.
Output
For each test case, output the answer as a value of a rational number modulo 109+7. Formally, it is guaranteed that under given constraints the probability is always a rational number pq (p and q are integer and coprime, q is positive), such that q is not divisible by 109+7. Output such integer a between 0 and 109+6 that p−aq is divisible by 109+7.