Trying to change fan speed but confused

I'm new with all of this and I'm trying to use Arduino to change the speed of a fan that is supplied by USB, I couldn't find a sketch for increasing the motor's speed.. also I'm not sure how to connect it to Arduino, this motor has two wire slots for each pole..

I appreciate any help!

Are you thinking in terms of chopping the existing cable off, providing some new power supply with a transistor or similar in the circuit that the Arduino can control with PWM to control the fan’s speed?

I was thinking about keeping the power supply for the fan as it is (supply from laptop), and including Arduino in the circuit somehow to increase the speed (and Arduino with its own power supply)… I’m not sure if I need other devices like a transistor or whether the power supply from laptop could work… I would appreciate your help!