| [Ken Arai - COMPLEX](https://soundcloud.com/diatomichail2/complex) | |
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| This is the hard version of the problem. In this version, the constraints on $n$ and the time limit are higher. You can make hacks only if both versions of the problem are solved. | |
| A set of (closed) segments is complex if it can be partitioned into some subsets such that | |
| * all the subsets have the same size; and * a pair of segments intersects if and only if the two segments are in the same subset. | |
| You are given $n$ segments $[l_1, r_1], [l_2, r_2], \ldots, [l_n, r_n]$. Find the maximum size of a complex subset of these segments. | |
| Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 10^3$). The description of the test cases follows. | |
| The first line of each test case contains a single integer $n$ ($1 \le n \le 3 \cdot 10^5$) — the number of segments. | |
| The second line of each test case contains $n$ integers $l_1, l_2, \ldots, l_n$ ($1 \le l_i \le 2n$) — the left endpoints of the segments. | |
| The third line of each test case contains $n$ integers $r_1, r_2, \ldots, r_n$ ($l_i \leq r_i \le 2n$) — the right endpoints of the segments. | |
| It is guaranteed that the sum of $n$ over all test cases does not exceed $3 \cdot 10^5$. | |
| For each test case, output a single integer: the maximum size of a complex subset of the given segments. | |
| In the first test case, all pairs of segments intersect, therefore it is optimal to form a single group containing all of the three segments. | |
| In the second test case, there is no valid partition for all of the five segments. A valid partition with four segments is the following: $\\{\\{ [1, 5], [2, 4] \\}, \\{ [6, 9], [8, 10] \\}\\}$. | |
| In the third test case, it is optimal to make a single group containing all the segments except the second. |