### Godot version

4.2-rc1

### System information

Windows 11

### Issue description

The issue does not exist on Godot 4.1.3, only in 4.2

When you call `seek()` on the AnimationPlayer, the animation does not update instantly and freezes for a brief moment.
So if you need to switch multiple animations in a short amount of time while also calling `seek()`, the animation will just stop.

### Steps to reproduce

- Add a Sprite2D
- Set a spritesheet for the Sprite2D
- Add an AnimationPlayer
- Create a Property animation track and add some frames
- Loop the animation
- Create a different animation for the same sprite an loop it
- Switch between animations and call `seek()` when you do

### Minimal reproduction project

The included project demonstrates the issue perfectly (I know that calling `play()` and `seek()` on each `_process()` is a bit overkill, but it does allow for a better demonstration). The same project works perfectly in Godot 4.1.3.

[AnimationBug.zip](https://github.com/godotengine/godot/files/13400011/AnimationBug.zip)

List<PropertyInfo> anim_names;
	for (const KeyValue<StringName, AnimationData> &E : animation_set) {
		HashMap<StringName, StringName>::ConstIterator F = animation_next_set.find(E.key);
		if (F && F->value != StringName()) {
			anim_names.push_back(PropertyInfo(Variant::STRING, "next/" + String(E.key), PROPERTY_HINT_NONE, "", PROPERTY_USAGE_NO_EDITOR | PROPERTY_USAGE_INTERNAL));
		}
	}
	for (const PropertyInfo &E : anim_names) {
		p_list->push_back(E);
	}
	p_list->push_back(PropertyInfo(Variant::ARRAY, "blend_times", PROPERTY_HINT_NONE, "", PROPERTY_USAGE_NO_EDITOR | PROPERTY_USAGE_INTERNAL));
}
void AnimationPlayer::_notification(int p_what) {