### 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 (playback.current.from);
}
double AnimationPlayer::get_current_animation_position() const {
	ERR_FAIL_NULL_V_MSG(playback.current.from, 0, "AnimationPlayer has no current animation.");
	return playback.current.pos;
}
double AnimationPlayer::get_current_animation_length() const {
	ERR_FAIL_NULL_V_MSG(playback.current.from, 0, "AnimationPlayer has no current animation.");
	return playback.current.from->animation->get_length();
}
void AnimationPlayer::set_autoplay(const String &p_name) {
	if (is_inside_tree() && !Engine::get_singleton()->is_editor_hint()) {
		WARN_PRINT("Setting autoplay after the node has been added to the scene has no effect.");
	}
	autoplay = p_name;
}
String AnimationPlayer::get_autoplay() const {