### 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 (is_playing() ? playback.assigned : "");
}
void AnimationPlayer::set_assigned_animation(const String &p_animation) {
	if (is_playing()) {
		float speed = playback.current.speed_scale;
		play(p_animation, -1.0, speed, signbit(speed));
	} else {
		ERR_FAIL_COND_MSG(!animation_set.has(p_animation), vformat("Animation not found: %s.", p_animation));
		playback.current.pos = 0;
		playback.current.from = &animation_set[p_animation];
		playback.assigned = p_animation;
		emit_signal(SNAME("current_animation_changed"), playback.assigned);
	}
}
String AnimationPlayer::get_assigned_animation() const {
	return playback.assigned;
}
void AnimationPlayer::pause() {
	_stop_internal(false, false);
}
void AnimationPlayer::stop(bool p_keep_state) {
	_stop_internal(true, p_keep_state);
}
void AnimationPlayer::set_speed_scale(float p_speed) {
	speed_scale = p_speed;
}
float AnimationPlayer::get_speed_scale() const {