### 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)

ERR_FAIL_COND_MSG(!animation_set.has(p_animation), vformat("Animation not found: %s.", p_animation));
	animation_next_set[p_animation] = p_next;
}
StringName AnimationPlayer::animation_get_next(const StringName &p_animation) const {
	if (!animation_next_set.has(p_animation)) {
		return StringName();
	}
	return animation_next_set[p_animation];
}
void AnimationPlayer::set_default_blend_time(double p_default) {
	default_blend_time = p_default;
}
double AnimationPlayer::get_default_blend_time() const {
	return default_blend_time;
}
void AnimationPlayer::set_blend_time(const StringName &p_animation1, const StringName &p_animation2, double p_time) {
	ERR_FAIL_COND_MSG(!animation_set.has(p_animation1), vformat("Animation not found: %s.", p_animation1));
	ERR_FAIL_COND_MSG(!animation_set.has(p_animation2), vformat("Animation not found: %s.", p_animation2));
	ERR_FAIL_COND_MSG(p_time < 0, "Blend time cannot be smaller than 0.");
	BlendKey bk;
	bk.from = p_animation1;
	bk.to = p_animation2;
	if (p_time == 0) {
		blend_times.erase(bk);
	} else {
		blend_times[bk] = p_time;
	}
}
double AnimationPlayer::get_blend_time(const StringName &p_animation1, const StringName &p_animation2) const {