### 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)

return playing;
}
void AnimationPlayer::set_current_animation(const String &p_animation) {
	if (p_animation == "[stop]" || p_animation.is_empty()) {
		stop();
	} else if (!is_playing()) {
		play(p_animation);
	} else if (playback.assigned != p_animation) {
		float speed = playback.current.speed_scale;
		play(p_animation, -1.0, speed, signbit(speed));
	} else {
		
	}
}
String AnimationPlayer::get_current_animation() const {