Car that bogs down

I have a 2004 v6 3.9 Mustang and when at idle and slowly pressing down the throttle, it bogs down and seems to get caught around 1500 rpm and then picks back up and runs fine after it. However, when giving it more throttle it doesn’t seem to hesitate at all. Also, when the revving at idle and letting off the throttle, the car drops below 500 rpm and shakes really bad and almost turns off. When stopping at red lights it does the same thing. Also, sometimes when driving the car does not want to accelerate at all but once I finally get down on the throttle it runs fine. The video is a video of it dropping low and almost turning off after reviving at idle and the car in the background is not me and it is also raining in the video. It also has fairly new spark plugs and a clean air filter. If anyone knows what this could be then please let me know, thanks. https://youtu.be/mbWrjGmWdp4

When a fuel injected engine has an idle problem, the first thing to suspect is the Idle Air Control valve.

https://www.rockauto.com/en/catalog/ford,2004,mustang,3.9l+v6,1432947,fuel+&+air,idle+air+control+(iac)+valve,6072

Tester

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It might also be some sort of vacuum leak. That can make the a/f mixture too lean at idle, and when you open the throttle by pressing on the accelerator pedal that makes it even more lean, so the engine misfires & bogs down, might even stall. As you gradually go faster the vacuum leak becomes small in relationship to the amount of air going through the throttle valve, so it runs better the more the accelerator pedal is pressed, after it gets up to speed. The first thing to do is check for lean and other codes in the computer diagnostic memory. After that probably a fuel trim measurement. A common source of a big vacuum leak is the brake booster, so you could check to make sure that holds vacuum. And you could spray starter spray here and there in the engine compartment, if the idle rpm speeds up noticeably you’ll know you probably have a vacuum leak in that area. Read the post around this same posting time about the 2004 Buick , rough idling, braking problem for add’l info.