Excellent Arrays
Description
Let's call an integer array $a_1, a_2, \dots, a_n$ good if $a_i \neq i$ for each $i$.
Let $F(a)$ be the number of pairs $(i, j)$ ($1 \le i < j \le n$) such that $a_i + a_j = i + j$.
Let's say that an array $a_1, a_2, \dots, a_n$ is excellent if:
- $a$ is good;
- $l \le a_i \le r$ for each $i$;
- $F(a)$ is the maximum possible among all good arrays of size $n$.
Given $n$, $l$ and $r$, calculate the number of excellent arrays modulo $10^9 + 7$.
The first line contains a single integer $t$ ($1 \le t \le 1000$) — the number of test cases.
The first and only line of each test case contains three integers $n$, $l$, and $r$ ($2 \le n \le 2 \cdot 10^5$; $-10^9 \le l \le 1$; $n \le r \le 10^9$).
It's guaranteed that the sum of $n$ doesn't exceed $2 \cdot 10^5$.
For each test case, print the number of excellent arrays modulo $10^9 + 7$.
Input
The first line contains a single integer $t$ ($1 \le t \le 1000$) — the number of test cases.
The first and only line of each test case contains three integers $n$, $l$, and $r$ ($2 \le n \le 2 \cdot 10^5$; $-10^9 \le l \le 1$; $n \le r \le 10^9$).
It's guaranteed that the sum of $n$ doesn't exceed $2 \cdot 10^5$.
Output
For each test case, print the number of excellent arrays modulo $10^9 + 7$.
Samples
4
3 0 3
4 -3 5
42 -33 55
69 -42 146
4
10
143922563
698570404
Note
In the first test case, it can be proven that the maximum $F(a)$ among all good arrays $a$ is equal to $2$. The excellent arrays are:
- $[2, 1, 2]$;
- $[0, 3, 2]$;
- $[2, 3, 2]$;
- $[3, 0, 1]$.