SP34032 INTDSET - Chiaki With Intervals

Chiaki has a set $ A $ of $ n $ intervals, the $ i $ -th of them is $ [l_i, r_i] $ . She would like to know the number of such interval sets $ S \subset A $ : for every interval $ a \in A $ which is not in $ S $ , there exists at least one interval $ b $ in $ S $ which has non-empty intersection with $ a $ . As this number may be very large, Chiaki is only interested in its remainder modulo $ (10^9+7) $ . Interval $ a $ has intersection with interval $ b $ if there exists a real number $ x $ that $ l_a \le x \le r_a $ and $ l_b \le x \le r_b $ .

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 \le n \le 2 \times 10^5 $ ) -- the number of intervals. Each of the following $ n $ lines contains two integers $ l_i $ and $ r_i $ ( $ 1 \le l_i < r_i \le 10^9 $ ) denoting the $ i $ -th interval. It is guaranteed that for every $ 1 \le i < j \le n $ , $ l_i \ne l_j $ or $ r_i \ne r_j $ and that the sum of $ n $ in all test cases does not exceed $ 2 \times 10^5 $ .

For each test case, output an integer denoting the answer.