HTML Block Conditional Logic Not Working

In continuing to build a form for my site, I am running into a small snag (or at least I hope it's a small snag, regarding the conditional logic of several HTML blocks. As you can view on this page, http://alltheyneedislove.org/adoption-application/, you will see 4 empty fieldsets. The logic works as so...The user clicks "No" for the Will this be your first pet question, then another field appears and asks them to select between 1 and 4 previous pets. Based on that number, the fieldsets with their associated fields would then appear. As it is right now, the fieldset stays, but the fields are working correctly.

You can see in this screenshot, how I have setup the logic, and its the same logic I have applied to the fields that work correctly: http://grab.by/kCjO

Any ideas on how to resolve this?

Ok, so I followed that posts instructions and even with the Twenty Eleven theme and only the gravity forms plugin activated, it is still failing to work properly. Again, I'm only trying to show the HTML Block that creates a fieldset if certain conditions are met. The code for the first HTML block is (to close the UL and start the first fieldset):

</li><!-- close the html field li -->
</ul><!-- close the list -->

<fieldset id="fieldset_1"><!-- open the first fieldset -->
<legend>Pet 1:</legend>
<ul id='gform_fields_4' class='gform_fields top_label description_below'><!-- reopen the list -->

The second block would be (to close the first fieldset and open a second):

</li><!-- close the html field li -->
</ul><!-- close the list -->
</fieldset><!-- close the first fieldset -->

<fieldset id="fieldset_2"><!-- open the second fieldset -->
<legend>Pet 2:</legend>
<ul id='gform_fields_4' class='gform_fields top_label description_below'><!-- reopen the list -->

I would repeat the second block for as many fieldsets as needed and then use this following code to close the last field set and return back to the normal gforms:

</li><!-- close the html field li -->
</ul><!-- close the list -->
</fieldset><!-- close the second fieldset -->

<ul id='gform_fields_3' class='gform_fields left_label description_below'>

At this point, I'm at a loss, since I believe I'm following the directions to a "T". Please help!

Regards,
Ben

Going through the validation link you provided, I cleaned up what I could, but it appears as though gravity forms is erroneously adding closing tags where they are not needed or because the HTML block themselves are closing off the list and gravity forms is trying to do the same in order to create the fieldsets. If i gave you access to the admin, can you take a look to see what I might be doing wrong?