It is the hard version of the problem. The only difference is that in this version $a_i \le 10^9$.

You are given an array of $n$ integers $a_0, a_1, a_2, \ldots a_{n - 1}$. Bryap wants to find the longest beautiful subsequence in the array.

An array $b = [b_0, b_1, \ldots, b_{m-1}]$, where $0 \le b_0 \lt b_1 \lt \ldots \lt b_{m - 1} \lt n$, is a subsequence of length $m$ of the array $a$.

Subsequence $b = [b_0, b_1, \ldots, b_{m-1}]$ of length $m$ is called beautiful, if the following condition holds:

• For any $p$ ($0 \le p \lt m - 1$) holds: $a_{b_p} \oplus b_{p+1} \lt a_{b_{p+1}} \oplus b_p$.

Here $a \oplus b$ denotes the bitwise XOR of $a$ and $b$. For example, $2 \oplus 4 = 6$ and $3 \oplus 1=2$.

Bryap is a simple person so he only wants to know the length of the longest such subsequence. Help Bryap and find the answer to his question.

Input

The first line contains a single integer $t$ ($1 \leq t \leq 10^5$)  — the number of test cases. The description of the test cases follows.

The first line of each test case contains a single integer $n$ ($2 \leq n \leq 3 \cdot 10^5$) — the length of the array.

The second line of each test case contains $n$ integers $a_0,a_1,...,a_{n-1}$ ($0 \leq a_i \leq 10^9$) — the elements of the array.

It is guaranteed that the sum of $n$ over all test cases does not exceed $3 \cdot 10^5$.

Output

For each test case print a single integer — the length of the longest beautiful subsequence.

Note

In the first test case, we can pick the whole array as a beautiful subsequence because $1 \oplus 1 \lt 2 \oplus 0$.

In the second test case, we can pick elements with indexes $1$, $2$ and $4$ (in $0$ indexation). For this elements holds: $2 \oplus 2 \lt 4 \oplus 1$ and $4 \oplus 4 \lt 1 \oplus 2$.