The date of the first and third saturday of every month will change (obviously) every month and every year. I think the best way to accomplish what you are looking for is to compare the current date against a list (such as an XML document) that defines the specific dates you want to invoke the script.
I currently use this logic for determining holidays.
I attached a shot of how this looks in the script (date.bmp). Basically what I am doing is setting the current date, then comparing that to my predefined dates in my XML document (xml.bmp) and if the result is "True", then follow that logic (using an "If" step).
You'll need to update your date list every year, but it will also allow you to add additional dates to the list if you need to, without messing with the script. Conversely you can also remove dates if needed.
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