Use arduino to close circuit?

Hello, I just recently purchased an Arduino Nano and I was hoping to make something that can make a VERY quick left mouse click. I have an old mouse that I have soldered two wires to the contacts points and when the wires are pressed together there is a click. I hooked them both up to a push button switch and it works fine, but how can I make the arduino do it? I was hoping to make the click faster than at least 75ms. I have attached a picture with my push button tester and the mouse board. Thanks!
If the picture didn't show correctly here is the link: https://drive.google.com/file/d/1aQOA_71cM6-zqmAIK47wrhOv2g_5dDUy/view?usp=sharing

Check the voltage of the wires. If one is 5volts, then connect the other one to an arduino digital output and send it a 1. If less than 5 volts, then connect that other wire to one of the analog output pins and send it that voltage. I doubt that a mouse draws much current, but you may wanna check first by connecting a meter between those two sires and see how much current flows. If much more than 20mA, you may need to drive the "non-hot" wire using a FET and arduino pin.

oops what I just suggested won't work since you need to get some kind of ground from the mouse as well. You may need to use just a FET with one wire going in and one out and triggered by the arduino.

Would there be any that you can recommend?

Regardless of how you connect the nano up, you will still require a means to tell the nano WHEN to close the circuit. Would seem silly to use a button to tell the nano when to push the button, right?

It doesn't take much to drive something like that, so maybe a 2n7000.