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This is an hard version of the problem. The only difference between an easy and a hard version is the constraints on $$$a$$$, $$$b$$$, $$$c$$$ and $$$d$$$.

You are given $$$4$$$ positive integers $$$a$$$, $$$b$$$, $$$c$$$, $$$d$$$ with $$$a \lt c$$$ and $$$b \lt d$$$. Find any pair of numbers $$$x$$$ and $$$y$$$ that satisfies the following conditions:

$$$a \lt x \leq c$$$, $$$b \lt y \leq d$$$,
$$$x \cdot y$$$ is divisible by $$$a \cdot b$$$.

Note that required $$$x$$$ and $$$y$$$ may not exist.

Input

The first line of the input contains a single integer $$$t$$$ $$$(1 \leq t \leq 10$$$), the number of test cases.

The descriptions of the test cases follow.

The only line of each test case contains four integers $$$a$$$, $$$b$$$, $$$c$$$ and $$$d$$$ ($$$1 \leq a \lt c \leq 10^9$$$, $$$1 \leq b \lt d \leq 10^9$$$).

Divisible Numbers (hard version)

Examples